Can't get under 200 turns, even without warmongering and focusing on just 4 cities. Should I be spending my gold on city-states or RAs?
Work backwards from your goal for a moment. You need culture - a lot of culture. This means you need to generate a lot of culture-per-turn-per-city. You need Culture Multipliers.
1.) A keystone to accomplishing that goal is to put up Broadcast Towers.
2.) That means you need to tech to Radio as soon as possible.
That means at least part of your primary concern is to get enough beakers and Free Technologies to get to Radio A.S.A.P. You'll also want the Sistine Chapel for its multiplier. You also need to get the Hermitage up and running a soon as possible.
The Hermitage is kind of a sub-quest unto itself, as Vexing pointed out how efficient using Legalism to meet the Opera House requirements can be.
Beyond those core multipliers and the tech path they represent you need to make sure you get the Oracle. In the end that free policy saves you the entire cost of the last policy you'd otherwise buy - the most expensive one.
Then you can focus on static production of culture - Monuments, Temples, Monasteries, Wats, Stonehenge, and various other wonders.
City-States are part of the base Culture Production, and thus have a lower priority than rushing to Radio. Don't drop money on Cultural City States until you're past the point where a Research Agreement would deliver Radio any sooner.
Myself, I tend to horde Great Scientists and slow-play my Oxford University so that I can just burn through at least Biology, Steam Power, Electricity, and Radio all in one turn. Once I'm sure I'll have Military Science and Scientific Method done in under 25 turns and I have 4 free tech to back them up I stop doing Research Agreements and switch to City States.
- Marty Lund