  The quartic scalar coupling $\lambda_5$ term, which violates the
lepton-number by two units in the Ma-model, is phenomenologically small when
the model is applied to the lepton-flavor violation (LFV) processes. In order
to dynamically generate the $\lambda_5$ parameter through quantum loop effects
and retain the dark matter (DM) candidate, we extend the Ma-model by adding a
$Z_2$-odd vector-like lepton doublet and a $Z_2$-even Majorana singlet. With
the new couplings to the Higgs and gauge bosons, the observed DM relic density
can be explained when the upper limits from the DM-nucleon scattering cross
sections are satisfied. In addition to the neutrino data and LFV constraints,
it is found that the DM relic density can significantly exclude the free
parameter space. Nevertheless, the resulting muon $g-2$ mediated by the inert
charged-Higgs can fit the $4.2\sigma$ deviation between the experimental
measurement and the SM result, and the branching ratio for $\tau\to \mu \gamma$
can be as large as the current upper limit when the rare $\mu\to (e \gamma, 3
e)$ decays are suppressed. In addition, it is found that the resulting $BR(\tau
\to \mu \rho)$ can reach the sensitivity of Belle II with an integrated
luminosity of 50 $ab^{-1}$.
