Hrmm, not sure if you already played the round, in which case this reply is not too useful. The GL is good because it doesn't obselete with Astronomy, which is pretty much a must get. Instead it obseletes with Corporation, which you really don't need unless you think the game is going until infantry.
Although Great Lighthouse is not worth much early on, it's still the equivalent of researching Currency, pretty much, in that the 2 penny ante trade routes spread over 7-8 cities is still a decent 20ish commerce.
When you get Astronomy, as long as you've kept your coastal cities somewhat large (> size 6), the effect of the trade routes will explode. With Currency + GL + base, you get 4 trade routes. If you open borders with multiple countries, that will usually amount to
5-6 commerce per trade route, often better than what the city itself is pulling in from tiles.
However however, GL is not ideal for an early cultural victory, in my opinion. The point at which you meet other civilizations is also around the period where you want to start thinking of jacking the culture slider, so all of those trade routes will go for naught. In fact colossus would be ideal, since it obseletes around when you don't need it anymore, and gives you a boost during the critical medieval period where you need to self research quite a heavy load.
Some other things to consider when going for CV:
Where to stop:
A good military tech would be a pretty good choice. You have to dig at least 3-4 techs past liberalism to find something that is useful culturally (broadcast towers, democracy civics, biology). Honestly, after you grab liberalism -> nationalism, you have all the pieces you really need (Hermitage, all the religions you can get, nationhood for emergency/precautionary drafts, free speech, free religion so less people hate you). If it seems like you are going fast (win in less than 60-80 turns when you flip the switch), I would just grab gunpowder, draft the hell out of all your non legendary/non-GP farm cities, turn off research and pray. If it's going to take longer, you might want to go to a more substantial military tech like chem/rifling (too bad you dont have horses
).
Which Three?
The good news is, your capital can support cottages on all the flats in its fat cross even without working the pig (windmill the hills after popping the slider).
The bad news is, you don't really have strong candidates for #2 and #3. Here are the other cities in order of expected base culture per turn after slider flip.
Red 1:
2-3 Ancient wonders: ~25 cpt
~11 cottages = ~ 66 cpt
4 windmills = ~6 cpt
Total ~ 97 cpt
Grey 4:
~13 (riverside) cottages (amortizing overlap with Red 1) = ~ 81 cpt
~5 water tiles = ~ 10 cpt
Total ~ 91 cpt
Grey 6:
6 cottages: ~ 36 cpt
8 water tiles: ~ 16 cpt
food surplus ~ 4 artists: ~ 24 cpt if you nab sistine
Total ~ 76 cpt
Blue 2:
~12 cottages = ~ 72 cpt
2 windmills = ~ 3 cpt
Total ~75 cpt
Green 3:
~15 food surplus = Run 7 artists ~42 cpt
NE would be in this city = ~ 8 cpt
Total ~50 cpt
These are obviously very rough estimates, but from the looks of it, I'd say Red 1 and Grey 4 have the best overall endgame cultural potential. Red 1 is hammer heavy, so you can chase some wonders, especially for times when you have nothing better to build. Grey 4 is very hammer light, so you might want to initially farm up the grass and whip in basic infrastructure and early temples/monastaries, and save the trees for chopping cathedrals. Basically, try not to waste any hammers there (markets/colliseum/courthouses etc are all very last priority. Obviously don't build any military buildings/units here).
So after these cities are cranking out culture, what should the other cities be doing to contribute?
Blue 2/Grey 7/ Grey 6/Grey 5 should farm up all its tiles (bulldozing cottages) and build/whip/draft military like it's going out of fashion. Also they can run merchant specialists so that you can maintain a high culture slider.
Green 3 should crank the artists hard under caste system. Only work your high food tiles.
Silverwhale city in the south should concentrate on getting its borders popped so you can access the two additional silvers. You're going to be running 100% culture, so happiness won't be a problem. However, health is an issue for capital and red 1. Ideally, try to trade your three silvers for rice/corn, which would give you 2 health a piece with granary, and either a banana or some gpt.
That would be my general plan. The devil is always in the details when playing a builder game though. Hope you pull it out!