OP would need a Chinese visa (about $140 for USA citizens) to enter China to get a French student visa? That sounds like a mess.

Plus, there will be lots and lots of Chinese students trying to get visas in China and I suspect that students in China rare scrutinized more than those in certain other countries. You probably wouldn't want to go through the same process as the Chinese students endure to get student visas. Also, I suspect that it could be very difficult to get an appointment with all of the Chinese students applying at this time of year.

If the OP must get the student visa elsewhere in Asia, I'd go for Singapore (where more English is spoken) or Japan, possibly Korea (another country where USA citizens can enter without visas).