Complexity of problem that can be solved by exponentially many SATs
The problem you defined is not in FNP as by definition the length of the answer should be polynomial in the length of its input.

Even if we are guaranteed that the length of "y" (output of the function) is polynomial in length of formula (i.e., there are at most polynomially many answer), this problem is not FNP except P= CoNP.

Proof:

$$\hspace{-1cm}  F(\phi)= y \Leftrightarrow  \forall A: $$
$$\hspace{5cm}                 {\huge(} A\models \phi \Rightarrow M(A,y) \land $$
$$\hspace{5cm}       A\not\models \phi \Rightarrow \lnot M(A,y){\huge )} $$

Where $M(A,y)$ is a polynomial time computable function returning true iff assignment $A$ is in the set of assignments $y$.

Definition of FNP requires us to have a polynomial time function $P$ given $\phi$ and $y$,  returns true iff $F(\phi)=y$. That is, deciding if the following formula is tautology or not should be verifiable in P:

$$\Phi (A)= {\huge(} \left(A\models \phi \Rightarrow M(A,y)\right) \land     \left(A\not\models \phi \Rightarrow \lnot M(A,y)\right){\huge )} $$