I think few non-invasive mods could be included in CBP 
UI - Religion Spread
This mod, courtesy of Whoward, introduces very useful and very cool feature: in Religion Overview there is fourth tab, Map, which features world map and colors cities' cultural borders by their majority religion. It looks awesome and, at least for me, was pretty helpful in religious strategy, as I can look where my religious enemies concentrate in one glance instead of checking every city on the map. And ReligionOverview.lua/xml are already changed by CBP so why not extend them with this useful feature?...
UI - Enhanced Demographics
Well I know CBP shouldn't be overloaded but this one mod blends into the game so perfectly you truly wonder why it wasn't added by Firaxis. Simply adds two more tabs to the Demographics' useless vanilla screen which normally offers only yours, best and worst score in each cathegory: first new tab has all statistical data from vanilla tab given for all civilisations in a handy UI and second one has ranking of biggest/most swimming in wonders cities in the world.
Krajzen's Great People Pack
Shameless advertisement, but this is an area untouched by CBP: not only the list of great people is greatly limited in-game, approximately 80% of them come from Western Europe (England, France, Germany, Italy, Rome, Greece)/USA
this mod, not conflicting with almost anything ever, adds 160 new names of great (research confirmed) people from all over the world:
It is available in download database. I'd also recommend my similar mods (also in database) adding great creators and their great works (99,9% bug-free confirmed, I spent much time checking that) from all over the world, including some 60s-80s musicians, but I am not sure about copyright. Well, roughly speaking over 3/4 of people I have found are dead anyway
As for other mods I think nice ones to include would be:
*Faster aircraft animations
*Better Polynesian City Names (workshop, "A main issue with the Polynesian City List in the Polynesian DLC was that it featured the names of Islands and Countries in Polynesia instead of actual cities and villages. This has been rectified with the Better Polynesian City Names mod, which changes the city names to actual cities, for example:
Samoa -> Apia
Tonga -> Nuku'alofa
Aotearoa -> Parihaka")
*Iroquian City Names (workshop, turns very awkward English names of local villages/tribes into correct ones)
*Hunnic City List (workshop, replaces ridiculous hunnic city list of Tlaxcala, Nankin and Timbuktu with accurate ones:
you need to admit this sounds much cooler (and obviously is more realistic
) than random collection of names from all continents.
*Whoward: City States No Allied Skirmishes - forces CS allied to one player to stop killing each other and make a truce; in vanilla civ5 there are often situations when CS1 was allied to player x, CS2 to player y and were fighting and will pointlessly fight once player z buys them.
*Whoward: Religious Settlers - settlers created in cities with religions can start a new city already with religion: if there are 4 citizens and 2 are muslim, 1 christian and 1 judaist, there is 50-25-25% chance of spawning settler which creates city whose first citizen is of appropriate faith.
*Whoward: Remove Ghost Routes - you can remove roads of dead civilisations.
*Whoward: Remove Jungle Gives Production - as for forests.
*Great Prophet Historical Names

UI - Religion Spread
This mod, courtesy of Whoward, introduces very useful and very cool feature: in Religion Overview there is fourth tab, Map, which features world map and colors cities' cultural borders by their majority religion. It looks awesome and, at least for me, was pretty helpful in religious strategy, as I can look where my religious enemies concentrate in one glance instead of checking every city on the map. And ReligionOverview.lua/xml are already changed by CBP so why not extend them with this useful feature?...
UI - Enhanced Demographics
Well I know CBP shouldn't be overloaded but this one mod blends into the game so perfectly you truly wonder why it wasn't added by Firaxis. Simply adds two more tabs to the Demographics' useless vanilla screen which normally offers only yours, best and worst score in each cathegory: first new tab has all statistical data from vanilla tab given for all civilisations in a handy UI and second one has ranking of biggest/most swimming in wonders cities in the world.
Krajzen's Great People Pack
Shameless advertisement, but this is an area untouched by CBP: not only the list of great people is greatly limited in-game, approximately 80% of them come from Western Europe (England, France, Germany, Italy, Rome, Greece)/USA

Spoiler :
GENERALS
Jan Sobieski, Janos Hunyadi, Skandenberg, Kutuzow, Simon Bolivar, Abu Bakr, Norman Schwarzkopf, Tran Hung Dao, David IV of Georgia, Nguyen Hue, Saladin, Jan Žižka, Nurhaci, Stanislaw Koniecpolski, Alp Arslan, Charles Martel, Lautaro, Pyotr Bagration, Ranjit Singh, Samudragupta, Mihal Bravu, Ahmad Shah Durrani, Cochise, Crazy Horse, Mahmud of Ghazni, Robert Guiscard, Xu Da, Andreas Prokop, Jan Karol Chodkiewicz, Arminius, Tokugawa Ieyasu, Piye, Naresuan, Josip Broz Tito, Ran Min, Huayna Capac, Cyrus the Great, Toussaint L'Ouverture, Gelawdewos, Nyazinga, Sundiata, K'Abel, Trung Vuong, Gwaggaeto the Great, Kanishka the Great, Oz Beg Khan, Amina, Malik Akbar, Hayam Wuruk,
SCIENTISTS
Brahmagupta,Omar Chajjam, al - Shirazi, Ulug Beg, al - Marwazi, Al Battani, Ahmes, al - Haytham, Janos Bolyali, Leó Szilárd, Baudhayan, Seki Takakazu, Madhawa, Florence Nightingale, Lagadha, Yajnavalkya, Panini, Chang Tshang, Al-Biruni , Qin Jiushao, Sofia Vasilyevna, Ramanujan, Stefan Banach, Shiing-Shen Chern, Alexey Postnikov, Pei Xiu, Xu Xiake, Shirley Jackson , Chien-Shiung Wu, Aziz Ab'Sáber, Zofia Kielan-Jaworowska, Juan Ignacio Molina
Ramachandran, Chandrasekhar, Merit-Ptah, Ahmed Zewail
GREAT ENGINEERS
Su Song, Carlos Paz de Araújo, al-Jazari, ibn Ma'ruf, Visvesvaraya, Abdus Salam, Patricia Bath, Yu Hao, Senenmut, Muzharul Islam, Ken Yeang, Sayyed Ali Shah, al-Isfahani, Ghiyathuddin, Krikor Balyan, Koca Kasım Ağa, Kálmán Tihanyi, Mimar Sinan, Anthemius of Tralles, Ilya Prigogine, Ximen Bao, Mukesh Ambani, Kankou Musou, Maslama al-Majriti Tapputi, Reza Ghadiri, Gunadharma, ibn Luqman
GREAT MERCHANTS
Muhammad Yunus, Taiichi Ohno, Jerry Yang, Carlos Slim, Li Ka-shing, Ibn Khaldun, Osman Ali Khan, Cosimo de Medici, Simón Iturri Patiño, Musa I of Mali, Ingvar Kamprad, Tenkaminen, Al-Mustansir, Suryavarman, Nyatsimba Mutota, Shah Abbas, Thomas Gresham, Yodoya Tatsugoro, Osei Tutu, Nathan Rothschild, Howqua, Cecil Rhodes, T.V. Soong,Tippu Tip, Rumaliza, Adriaan van der Hoop, Tanaka Shōsuke, Scrooge McDuck
GREAT ADMIRALS
Cornelis Tromp, George of Antioch, Kanhoji Angre, Keumalahayati, Gustav Horn, Pytheas, Xu Fu, Zhang Qian, Wang Dayuan, Hong Bao, Hannu, Ngo Quyen
Jan Sobieski, Janos Hunyadi, Skandenberg, Kutuzow, Simon Bolivar, Abu Bakr, Norman Schwarzkopf, Tran Hung Dao, David IV of Georgia, Nguyen Hue, Saladin, Jan Žižka, Nurhaci, Stanislaw Koniecpolski, Alp Arslan, Charles Martel, Lautaro, Pyotr Bagration, Ranjit Singh, Samudragupta, Mihal Bravu, Ahmad Shah Durrani, Cochise, Crazy Horse, Mahmud of Ghazni, Robert Guiscard, Xu Da, Andreas Prokop, Jan Karol Chodkiewicz, Arminius, Tokugawa Ieyasu, Piye, Naresuan, Josip Broz Tito, Ran Min, Huayna Capac, Cyrus the Great, Toussaint L'Ouverture, Gelawdewos, Nyazinga, Sundiata, K'Abel, Trung Vuong, Gwaggaeto the Great, Kanishka the Great, Oz Beg Khan, Amina, Malik Akbar, Hayam Wuruk,
SCIENTISTS
Brahmagupta,Omar Chajjam, al - Shirazi, Ulug Beg, al - Marwazi, Al Battani, Ahmes, al - Haytham, Janos Bolyali, Leó Szilárd, Baudhayan, Seki Takakazu, Madhawa, Florence Nightingale, Lagadha, Yajnavalkya, Panini, Chang Tshang, Al-Biruni , Qin Jiushao, Sofia Vasilyevna, Ramanujan, Stefan Banach, Shiing-Shen Chern, Alexey Postnikov, Pei Xiu, Xu Xiake, Shirley Jackson , Chien-Shiung Wu, Aziz Ab'Sáber, Zofia Kielan-Jaworowska, Juan Ignacio Molina
Ramachandran, Chandrasekhar, Merit-Ptah, Ahmed Zewail
GREAT ENGINEERS
Su Song, Carlos Paz de Araújo, al-Jazari, ibn Ma'ruf, Visvesvaraya, Abdus Salam, Patricia Bath, Yu Hao, Senenmut, Muzharul Islam, Ken Yeang, Sayyed Ali Shah, al-Isfahani, Ghiyathuddin, Krikor Balyan, Koca Kasım Ağa, Kálmán Tihanyi, Mimar Sinan, Anthemius of Tralles, Ilya Prigogine, Ximen Bao, Mukesh Ambani, Kankou Musou, Maslama al-Majriti Tapputi, Reza Ghadiri, Gunadharma, ibn Luqman
GREAT MERCHANTS
Muhammad Yunus, Taiichi Ohno, Jerry Yang, Carlos Slim, Li Ka-shing, Ibn Khaldun, Osman Ali Khan, Cosimo de Medici, Simón Iturri Patiño, Musa I of Mali, Ingvar Kamprad, Tenkaminen, Al-Mustansir, Suryavarman, Nyatsimba Mutota, Shah Abbas, Thomas Gresham, Yodoya Tatsugoro, Osei Tutu, Nathan Rothschild, Howqua, Cecil Rhodes, T.V. Soong,Tippu Tip, Rumaliza, Adriaan van der Hoop, Tanaka Shōsuke, Scrooge McDuck

GREAT ADMIRALS
Cornelis Tromp, George of Antioch, Kanhoji Angre, Keumalahayati, Gustav Horn, Pytheas, Xu Fu, Zhang Qian, Wang Dayuan, Hong Bao, Hannu, Ngo Quyen
It is available in download database. I'd also recommend my similar mods (also in database) adding great creators and their great works (99,9% bug-free confirmed, I spent much time checking that) from all over the world, including some 60s-80s musicians, but I am not sure about copyright. Well, roughly speaking over 3/4 of people I have found are dead anyway

Spoiler :
Writers:
Stanislaw Lem, Wislawa Szymborska, Zbigniew Herbert, Jaroslav Hasek, Ferenc Molnar, Ismail Kadare, Vaclav Havel, Marko Miljanov, Masiela Lusha, Tove Jansson, Henrik Ibsen, Johann Huizinga, Dafydd ap Gwilym, Ali Bin abu - Talib, Aneirin, Cassiodorus, Li Bai, T'ao Ch'ien, Bai Juyi, Ts'ao Sung, Saadi, Attar of Nishapur, Jalal ad - Din Rumi, Archipoeta, Ippen, Kabir, Ihara Saikaku, Nazim Hikmet, Ataol Behramoglu, Khakheperresenb, Ptanhotep, Thiruvalluvar, Rabindranath Tagore, Nhat Hanh, Abdul Ghani Khan, Nadine Gordimer, Chinua Achebe, Frank Chipasula, Wole Soyinka, Giannina Braschi, Vatsyayana, Gioconda Belli, Gabriel Garcia Marquez, Lao Tzu, Orhan Pamluk, Muhammed Fuzuli, Gabriela Mistral, Romulo Gallegos, Julio Cortazar, Jorge Luis Borges, Octavio Paz, Ah Bam, Francisco Arana Xajila, Karel Capek, Jnanappana, Nitlacoya, Naryan, Chingiz Aitmatov, Mwana Kupona, Heo Nanseolheon, Lady Ise, Christine de Pizan, Celestine Hitiura Vaite, Griot Djeli Mamadou Kounyate, Zera Yacob, Bessie Head, Mwengo, CHimamanda Ngozi Adichie, Nnedi Okorafor, Ho Xuan Huong, Preah Bothumthera Som, Peter Watts, N. Scott Momaday, Henry Longfellow, Joy Harjo, Sherman Alexie, Leslie Marmon Silko, Sunthorn Phu, Mpu Prapanca, Nguyen Du, Wu Cheng'en, Luo Guazhong, Anna Frank, Deganawidah, Iryeon and FINALLY 'Tibet' (Epic of King Gesar, Tibetan national epic, no known single author) and 'The Horseman' (Secret History of Mongols)
Musicians:
Abraham Zevi Idelsohn, Adele, Amel Brahim, Antonio Carlos Yobim, 'Apache' (tribal music), Aram Khachaturian, Archilocos (ancient Greece), Astor Piazzolla, A.T. Mahmud, Aziza Mustafa Zadeh, Bob Dylan, Bob Marley, Cesaria Evora, Chen Yi, Christopher Tin (Baba Yetu), Coolio, Cartola, Dede Efendi, Erhu (ancient China), Fairouz, Fela Kuti, Frank Sinatra, Gregory Isaacs, Guns'n'Roses, Hans Zimmer, Henryk Wieniawski, Hildegard von Bingen (medieval), Ibrahim Ferrer, Iron Maiden, Isihija, Israel Kamakawiwo'ole, Itzhak Perlman, Joanne Shenandoah, Jose Mauricio Nunes Garcia, Kaoma, Led Zeppelin, Louis Armstrong, Marilyn Monroe, Metallica, Michael Jackson, Mohicans (tribal music), Mr Phoon Yew Tien, Mulatu Astatke, Nat Yontararak, Nico Vega, PSY, Queen, Ravi Shankar Raga, Romica Puceanu, Sa Ding Ding, Saint Kassia (Byzantium), Scott Joplin, Stevie Wonder, The Beatles, The Rolling Stones, Xian Xinghai, Yiruma, Ylvis, Yoko Kanno, Yo-Yo Ma.
Stanislaw Lem, Wislawa Szymborska, Zbigniew Herbert, Jaroslav Hasek, Ferenc Molnar, Ismail Kadare, Vaclav Havel, Marko Miljanov, Masiela Lusha, Tove Jansson, Henrik Ibsen, Johann Huizinga, Dafydd ap Gwilym, Ali Bin abu - Talib, Aneirin, Cassiodorus, Li Bai, T'ao Ch'ien, Bai Juyi, Ts'ao Sung, Saadi, Attar of Nishapur, Jalal ad - Din Rumi, Archipoeta, Ippen, Kabir, Ihara Saikaku, Nazim Hikmet, Ataol Behramoglu, Khakheperresenb, Ptanhotep, Thiruvalluvar, Rabindranath Tagore, Nhat Hanh, Abdul Ghani Khan, Nadine Gordimer, Chinua Achebe, Frank Chipasula, Wole Soyinka, Giannina Braschi, Vatsyayana, Gioconda Belli, Gabriel Garcia Marquez, Lao Tzu, Orhan Pamluk, Muhammed Fuzuli, Gabriela Mistral, Romulo Gallegos, Julio Cortazar, Jorge Luis Borges, Octavio Paz, Ah Bam, Francisco Arana Xajila, Karel Capek, Jnanappana, Nitlacoya, Naryan, Chingiz Aitmatov, Mwana Kupona, Heo Nanseolheon, Lady Ise, Christine de Pizan, Celestine Hitiura Vaite, Griot Djeli Mamadou Kounyate, Zera Yacob, Bessie Head, Mwengo, CHimamanda Ngozi Adichie, Nnedi Okorafor, Ho Xuan Huong, Preah Bothumthera Som, Peter Watts, N. Scott Momaday, Henry Longfellow, Joy Harjo, Sherman Alexie, Leslie Marmon Silko, Sunthorn Phu, Mpu Prapanca, Nguyen Du, Wu Cheng'en, Luo Guazhong, Anna Frank, Deganawidah, Iryeon and FINALLY 'Tibet' (Epic of King Gesar, Tibetan national epic, no known single author) and 'The Horseman' (Secret History of Mongols)
Musicians:
Abraham Zevi Idelsohn, Adele, Amel Brahim, Antonio Carlos Yobim, 'Apache' (tribal music), Aram Khachaturian, Archilocos (ancient Greece), Astor Piazzolla, A.T. Mahmud, Aziza Mustafa Zadeh, Bob Dylan, Bob Marley, Cesaria Evora, Chen Yi, Christopher Tin (Baba Yetu), Coolio, Cartola, Dede Efendi, Erhu (ancient China), Fairouz, Fela Kuti, Frank Sinatra, Gregory Isaacs, Guns'n'Roses, Hans Zimmer, Henryk Wieniawski, Hildegard von Bingen (medieval), Ibrahim Ferrer, Iron Maiden, Isihija, Israel Kamakawiwo'ole, Itzhak Perlman, Joanne Shenandoah, Jose Mauricio Nunes Garcia, Kaoma, Led Zeppelin, Louis Armstrong, Marilyn Monroe, Metallica, Michael Jackson, Mohicans (tribal music), Mr Phoon Yew Tien, Mulatu Astatke, Nat Yontararak, Nico Vega, PSY, Queen, Ravi Shankar Raga, Romica Puceanu, Sa Ding Ding, Saint Kassia (Byzantium), Scott Joplin, Stevie Wonder, The Beatles, The Rolling Stones, Xian Xinghai, Yiruma, Ylvis, Yoko Kanno, Yo-Yo Ma.
As for other mods I think nice ones to include would be:
*Faster aircraft animations
*Better Polynesian City Names (workshop, "A main issue with the Polynesian City List in the Polynesian DLC was that it featured the names of Islands and Countries in Polynesia instead of actual cities and villages. This has been rectified with the Better Polynesian City Names mod, which changes the city names to actual cities, for example:
Samoa -> Apia
Tonga -> Nuku'alofa
Aotearoa -> Parihaka")
*Iroquian City Names (workshop, turns very awkward English names of local villages/tribes into correct ones)
*Hunnic City List (workshop, replaces ridiculous hunnic city list of Tlaxcala, Nankin and Timbuktu with accurate ones:
Spoiler :
Attila's Court
Tapiomarton
Segedin
Sikambruk
Turuk
Hunuk
Akatziruk
Kutrigur
Sabir
Utigur
Suvar
Kevi
Antes
Sklavenir
Bastarnuk
Suebir
Aduni
Jegi
Egerseg
Telikhi
Laiduni
Bastetir
Tesongadi
Dorgathegi
Khalatir
Suchongadi
Livi
Tapiomarton
Segedin
Sikambruk
Turuk
Hunuk
Akatziruk
Kutrigur
Sabir
Utigur
Suvar
Kevi
Antes
Sklavenir
Bastarnuk
Suebir
Aduni
Jegi
Egerseg
Telikhi
Laiduni
Bastetir
Tesongadi
Dorgathegi
Khalatir
Suchongadi
Livi
you need to admit this sounds much cooler (and obviously is more realistic

*Whoward: City States No Allied Skirmishes - forces CS allied to one player to stop killing each other and make a truce; in vanilla civ5 there are often situations when CS1 was allied to player x, CS2 to player y and were fighting and will pointlessly fight once player z buys them.
*Whoward: Religious Settlers - settlers created in cities with religions can start a new city already with religion: if there are 4 citizens and 2 are muslim, 1 christian and 1 judaist, there is 50-25-25% chance of spawning settler which creates city whose first citizen is of appropriate faith.
*Whoward: Remove Ghost Routes - you can remove roads of dead civilisations.
*Whoward: Remove Jungle Gives Production - as for forests.
*Great Prophet Historical Names