Is there a known way to convert any $QBF_2$-formula into an equisatisfiable $QBF_2$-formula in CNF in polynomial time?
Yes, for exactly the reasons you listed.
I don't know of any way to do what you want.  However, it is possible to convert it to a formula of the form $\neg \forall x \exists y \Psi$ (negate it, then convert $\neg \Phi$ to CNF via the Tseitin transform and let $\Psi$ be the result), so if you have an oracle that will solve QBF2-CNF queries, it can be used to solve QBF2 queries.