The anomalous results of several recent $b\to s\mu^+\mu^-$ measurements could be initial evidence of new physics beyond the standard model in $b\to s$ transitions. Supposing this to be the case, we consider a scenario in which a heavy $Z'$ boson is responsible for the anomalies. We further assume that its interactions also influence the rare nonleptonic decays of the $B_s$ meson which tend to be dominated by electroweak-penguin contributions and are purely isospin-violating. Most of these $B_s$ decay modes are not yet observed, and their rates are expected to be relatively small in the standard model. Taking into account various constraints, we find that the $Z'$ effects can enhance the rates of some of the decays, particularly $\bar B_s\to\eta\pi^0,\phi\pi^0$, by up to an order of magnitude above their standard-model predictions. This $Z'$ scenario is therefore potentially testable in upcoming experiments at LHCb.