I have to admit that I don't generally go for the Cultural-VCs, so I can't offer any specific strategies*. But some lessons are universal...
(As I might have mentioned before!) If you (general 'you', here) pick a preferred VC, but don't then actively direct your Civ's development primarily towards that goal, you may well be setting yourself up to get a nasty surprise — like being pipped for a Diplo-loss when you wanted Space, or a Dom-loss while you were hoping for 20K, or actually losing Culturally, as described in
another recent post (unlike the others, I don't remember this last
ever happening in any of my games, so I've no idea how
@strongreaction 'managed' it!).
So taking a 'consolation prize' approach to playing Civ3 is not generally ideal,
especially for the Cultural VCs, which (I've been told) you need to start prepping for as early as possible — not least because (1) for a 100K-win, you need to get there before the next-best Civ gets 50K, and (2) once an Improvement/ Wonder's been up for 1000 years (i.e. 20-25 turns for much of the BCs), its Cultural-output gets doubled — so the earlier you build them, the better.
That way, a Culture-focussed game (on a Standard-size map, anyway) then might not even go as far as Fission/UN, especially not at Chieftain, where you can build/research everything twice as fast as any one of the AI-Civs — and if it does, it should only be because
you got there (long) before they did. As regards 'losing' Bachs and Shakes, while the AI-Civs
do tend to prioritise the upper Education/Wonder-branch of the Mid-Age tech-tree, even at Emp they're often quite tardy with going for MusicTheory (and FreeArt), so you should easily be able to snag
both those GWs long before a Diplo-loss (in the Modern Age) is even possible — especially at Chieftain.
But the main trick for any Wonder-race is really to start prebuilding while you're still researching towards the Wonder-tech: ideally, using a shield-bank that will fill shortly after you get the tech, so you can switch over and finish the Wonder off quickly, rather than waiting until you have the tech before you even start building it.
*(other than, I've read that the Byzzies are good for the 20K because they can build the cheap AA SEA-Wonders in Constantinople; and the Celts are good for 100K by whipping newly grown Agricultural citizens into half-price Temples/Cathedrals in Feudal ICS'd city-pox)