Division by two of functions in #P
I try to give my intuition why I think this is unlikely to hold. Take your favorite problem in $PPA$, and convert it into a problem in $\sharp P$, e.g., our function $f$ can be the number of Hamiltonian cycles in an input 3-regular graph containing a certain fixed edge. From the parity argument we know that $f$ is always even, so you can define $F:=f/2$ and I see no reason why $F$ would be in $\sharp P$.