By extending the minimal scotogenic model with a spontaneously broken global symmetry $U(1)'$ and a preserved $\mathbb{Z}_2$ symmetry, we build a seesaw model for generating neutrino masses at three-loop level. The new particles have masses at the TeV scale and relatively large Yukawa couplings, which leads to sizable rates for charged lepton flavor violation processes, well within future experimental reach. The model is able to successfully explain the $W$ mass anomaly and provides a viable fermionic or scalar dark matter candidate, while satisfying all current constraints imposed by neutrinoless double-beta decay, charged-lepton flavor violation, and electroweak precision observables.