Assignment to make formula unsatisfiable
Your problem is the canonical $\Sigma_2^P$-complete problem:
$$
\exists \vec{S} \forall \vec{A} \lnot F(\vec{A},\vec{S}).
$$
As such, it is thought to be more difficult than SAT (which is $\Sigma_1^P$). Solving it with a few SAT-oracle calls is akin to solving SAT itself efficiently (the P vs. NP question), though it could be that $\Sigma_2^P = \Sigma_1^P$ while $P \neq NP$, so in some sense there is more hope for your problem than for SAT itself.