Separability of the nearest power of two function
No, $f(x+y)$ is not separable.

Consider: $f(3+5)=8$.  Also $f(3)=4$ and $f(5)=8$.

At the same time, $f(4+5)=16$.  Also $f(4)=4$ and $f(5)=8$.

It follows that $f(x+y)$ is not determined by $f(x)$ and $f(y)$ (consider $y=5$, $x_1=3$, $x_2=4$), so there is no function $h$ such that $f(x+y)=h(f(x),f(y))$.