For obvious reasons of magnitude, $(2^n -1)!! -1$ cannot divide $2^n$. On the other hand, $2^n$ divides $(2^n -1)!! -1$ for $n\geq 3$. It is a case of "generalized Wilson theorem", proved very much like the original Wilson theorem: all residues $a$ modulo $2^n$ such that $a^2\not\equiv 1\pmod n$ are split into pairs with product 1, and 8 square roots of 1 modulo $2^n$ have product 1.