The answer is no.
If $j$ maps each $V_{\omega+n+1}^M$ onto $V_{\omega+n}^M$ for $n\in\omega^M$, then $j[V_{\omega+\omega}^M]=V_{\omega+\omega}^M$ and so it cannot be that $j[V_{\omega+\omega+1}^M]=V_{\omega+\omega}^M$ also, if $j$ is one-to-one, since all the elements in the target are already used.