Can't you say "You're Chinese but speak French?" It wasn't accepted when I wrote that but it's acceptable in English to say that
I have a question about the tense of the verb:
As "biliyorsun" is used, wouldn't it mean "speaking French (right now)"?
So shouldn't it be "bilirsin"?
This is more of an idiomatic thing in Turkish and English. In English, we normally say "I speak X" to mean we know the language. Turkish just uses the verb "to know" for this. Turkish also uses this verb in the present continuous tense to show this (just a thing that is done, no real reason per say).
If say you are speaking a language right now, you use the verb "to speak" in Turkish (Ben Xçe konuşuyorum).
In this example it would be good to distinguish between 'but your mother language is French' and 'but you speak French (as well / as a second language)'. How can you do that in Turkish?