Can a commutative block cipher be indistinguishable from a random permutation, for fixed key?
It can't be achieved under the assumptions you are making, because the attacker can distinguish it by selecting an arbitrary $k'$, and checking if $E(k')$ commutes with the permutation in question.  That is, to check a permutation $P$, we pick an arbitrary $x$, and check if:

$E(k', P(x)) = P(E(k',x))$

This equation always holds if $P = E(k)$ for some value $k$, and rarely holds if $P$ is a random permutation.