  Recently, the experimental values of the muon $(g-2)_\mu$ and of the $W$
boson mass $m_{_W}$ have both indicated significant deviations from the SM
predictions, motivating the exploration of extensions with extra particles and
symmetries. We revisit a lepton portal model with $U(1)'$ gauge symmetry where
an extra Higgs doublet, a scalar singlet and one $SU(2)_L$ singlet vector-like
fermion are introduced. In this model, $(g-2)_\mu$ can be explained by extra
one-loop contributions from the vector-like lepton and the $Z'$ boson, whereas
$m_{_W}$ can be increased by a tree-level mixing between the $Z$ and $Z'$.
Setting the $Z'$ and lepton couplings at low energies to account for the SM
anomalies, we perform a Renormalization Group analysis to investigate on the
high-energy behaviour of the model, in particular on the issue of vacuum
stability. We find that in the alignment limit for the two Higgs doublets, the
Landau pole and the scale where perturbativity is lost are of order
$10-100\,{\rm TeV}$, not far from the scales experimentally reached so far, and
sensibly lower than the stability scale. We show how the Landau pole can be
increased up to $\sim10^9\,{\rm GeV}$ in a misaligned scenario where the
experimental anomalies are still accommodated and a positive shift of the Higgs
quartic coupling to improve stability can be achieved.
