Does exist NP language that is Cook Levin deterministic reducible to xor satisfiability in polynomial time?
XORSAT is in P, so any such language would have to be in P as well.

As such, it is easy to exhibit such an example: the problem XORSAT itself satisfies all your requirements, as XORSAT is Cook-Levin reducible to XORSAT (let $f$ be the identity function).