What is the computational complexity of this SAT Variant
The class ${\bf C_=P}$ is defined to be the set of languagges $L$ such that there exist functions $f \in {\bf \# P}$,$t\in {\bf FP} $ and for all $x, x\in L$ if and only if $f(x)=t(x)$.  Exact-SAT is the standard complete language for ${\bf C_=P}$. Exact-SAT = $\{F, k\} | F$ is a Boolean formula with exactly $k$ satisfying truth assignments.  It is known that ${\bf CoNP} \subseteq {\bf US} \subseteq {\bf C_=P}$  See the paper "On the Power of Deterministic Reductions to ${\bf C_=P}$ " by Frederic Green.