I think either cultural or space are possible. Domination/conquest are out, and I've never figured out diplomatic except as back-door domination. My concern about cultural is that you have to become completely helpless. Either HC or Shaka could ruin your game. And there's two folks you haven't met yet, who could be just as bad. I would shoot for the stars.
To do that, you need to trade your way back up. That means a deep beeline, probably to Democracy. Only HC even has Nationalism, so I think you can get ahead on that line, trading Constitution for things like Education and Printing Press. Plus, Democracy is just about the single most important tech for a cottage-based space race (for Emancipation to grow cottages; US is eventually nice for the production bonus, but not until much later).
Do not trade for junk like Archery! And Theology, for that matter. It's a dead end, providing only a civic that you're never going to use. I think you want Machinery (as a pre-req to Printing Press), and I agree with the suggestion to research it for a turn or two and then trade Philosophy for it. After you've done that, gift Philosophy to Shaka (or sell it for whatever cash he has) for the relations bonus. Then when you're ready to trade Nationalism or Constitution, he'll be willing.
You need to get cities down where you want them, before those spots disappear. Is there a good spot north of Lyons? I don't remember the more detailed maps. And take that barbarian city yourself. It's a good spot. Then cottages, cottages, cottages. Try to lay off the whip -- your happy caps are pretty high now, and you want to be working a lot of tiles. A lot of cottages, in particular. Courthouses, for example, may no longer be worth the population you have to burn to whip them. (Exception would be big French cities coming out of revolt into massive unhappiness. So you still get to whip some

.)
It won't be a cake-walk like the last space race, but I think this game is very winnable. You have a lot of land, which is the primary requirement for space races.
peace,
lilnev