To the best of my knowledge, FP^NP[1] is a class of problems solvable by a deterministic Turing machine with only 1 NP query passed through an oracle.
If it were possible to show that #SAT is in FP^NP[1] would that have any bearing on the relationship between NP-complete problems and FP^NP[1] problems?
Toda's theorem puts the whole polynomial hierarchy (PH) within $P^{\#P}$, which is another way of saying that all the problems in PH are readily reducible to counting problems. $P^{NP}$ (the decision problem variant of $FP^{NP}$) being as powerful as $P^{\#P}$ would therefore collapse the polynomial hierarchy to the second level. This result would suggest that NP-completeness and PSPACE-completeness are a whole lot closer in difficulty than thought currently, because an NP oracle plus a tiny bit of extra fiddling conquers the whole polynomial hierarchy.