I am not sure about how to handle the diplomacy side though. I don't want most of those to cause diplo relations to suffer.
Actually you need a good mechanism to make e.g. the Elves hate Mordor and vice-versa. Currently everyone is still too nicey-nicey. The Easterlings and Haradrim should be likely to buddy up to Mordor and should dislike/distrust Gondor. Cirdan and Sauron should be oil and water. Religion provides a good means to achieve this in most mods, so a mix of "good" and "evil" Values would give the mod some much-needed grist. To support this function the list of Values will need considerable rework.
Also, it's not entirely true that there are no religions in Middle Earth, more that Tolkien baked his own Catholic beliefs into his pre-Christian creation.
In Numenor, Sauron pushed Morgoth-worship and built a gigantic temple to it. No doubt he brought that handy idea back to Middle Earth to spread to his servants and those he wanted to recruit to his service. At its simplest you could have all evil realms worship Morgoth and have all Elvish realms revere the Valar, and set these two beliefs to be extremely averse to one another. You could then add other beliefs to nuance it. For example, the Numenoreans would have a general reverence for the West, with a diminished animosity to those who revere the Valar but a strong aversion to Morgoth Worshippers. Dwarves would revere Durin (all of those we meet in LotR are Durin's folk so although the other 6 kindreds may prefer their own Father, this is not an issue here) and would be moderately averse to all others, but more to evil due to their bad experiences with Orcs. Some Men even worshipped Sauron, so you can nuance evil beliefs with Sauron Worshippers. Better names could be devised; I've used descriptive names here for clarity.
As a sample, we could have (with, in each case, +3 to anyone who shares their own belief):
Valar Reverers (Elves): -1 to West Reverers, -1 to Durin's Folk, -4 to Morgoth/Sauron Worshippers
West Reverers (Numenoreans/Arnor/Gondor): -1 to Valar Reverers, -1 to Durin's Folk, -3 to Morgoth Worshippers, -4 to Sauron Worshippers
Durin's Folk (Dwarves): -1 to Valar/West Reverers, -4 to Morgoth/Sauron worshippers
Morgoth Worshippers (Mordor/Haradrim): -1 to Sauron Worshippers, -3 to West Reverers/Durin's Folk, -4 to Valar Reverers
Sauron Worshippers (Easterlings): -1 to Moroth Worshippers, -3 to Valar Reverers, -4 to West Reverers/Durin's Folk
Saruman might also come up with some religious notion (Isdengard/Dunlendngs); he stole a lot of his ideas from Sauron. You could also add a generic worship similar to, say, Druidism, for Dale and other generally unaligned realms (Dale specifically because of their co-operation with the Dwarves of the Lonely Mountain). That gives us 7 distinct Value systems.
Note these are not perfectly symmetrical - Dwarves can hate Morgoth Worshippers more Morgoth Sauron Worshippers hate Durin's Folk. This is reflective of their past history with Sauron. I think some civs (e.g. Spain in vanilla BtS and most mods) take their religion more seriously than others, so this may also be a way to tweak the effect.
Some mods assign Religions to specific regions (e.g. if you are the first to discover the tech that enables Hinduism but don't have a city in India, you don't found Hinduism). You could look at a similar mechanism to make civs based in certain areas more likely to go for a Value innate to that area. Or you could just assign a specific Value to each civ from the start.
Edited to add: There would indeed be no Holy Cities, but Valar/West Reverers could build Holy Places and Sauron/Morgoth Worshippers could build Temples; Dwarves would build Tombs. There would be no Monasteries as such but their research boost could be added to some other new building. Cathedrals would be Great versions of Holy Places and Temples. Priests are a problem but could be renamed and reoriented a bit as Seers and Great Seers with the same Coins + Hammer effect as Priests.