Examples of #P problems which are in FP ? Is there any case of SAT problems (except the affine ones) whose counting version can be computed in polynomial time ?
$SAT$ and $\#SAT$ for CNF formulas of bounded rank are tractable. See $[1]$.


Andrei S. et al. A hierarchy of tractable subclasses for SAT and counting
SAT problems. 11th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing.