China has it's client countries that it owns for the most part except for in name. Same with the USA.
Sometimes not even an incentive, outside of personal desire. People are willing to donate to things they believe are worthy causes - for many, space travel and exploration meets the criteria.
EDIT: Also I recall this other time we had a "peaceful two-government world", it was called the Cold War. :p
Not an exact analogy I know, but really, I don't see a ton of good coming from out of it even if you could do it completely peacefully; add in the terrible violence and destruction of war, and the ends do not justify the means, as they rarely do.
No I don't see much good coming from it either, and it's totally not necessary from here. But it would be a setback that would be cheered around the world. War does create innovation though. A lot of the best tech is used in war first.
Also I'm still holding out hope for starting my own seasteading libertarian nation. Not gonna happen in that world.
A peaceful 2 government world would bring very little innovation. Innovation comes from competition. All that's required for space travel is a free market and an incentive.