NASA astronauts get mental health evaluations, so yes.
On the other hand, if you have your own rocket, I don't think NASA could stop you. (Though perhaps there's an FAA qualification for being a pilot? But not for supercargo.)
On the third hand, I don't see why the US would ban a American astronaut, launching on a Chinese mission. (While the US "could" in principle, eg, by sending a cruise missile to destroy the launch facility, I don't think that's your question.)