Recovering a signal after nonuniform sampling
I agree with your result for $Z(j\omega)$. Apart from scaling, what you have is the original spectrum with positive and negative frequencies swapped and multiplied with factors $1-j$ and $1+j$, respectively. In order to restore the original signal, we need to add right-shifted and left-shifted versions of the spectrum, while getting rid of the complex factors. Final lowpass filtering removes the redundant components in the frequency range $[M,2M]$.
The necessary shifting and complex scaling is achieved by multiplication with
$$e^{jMt}(1+j)+e^{-jMt}(1-j)=2\cos(Mt)-2\sin(Mt)\tag{1}$$
Modulation with $(1)$, lowpass filtering with cut-off frequency $M$, and appropriate scaling restores the original signal.