We report on a study of the sensitivity of the MEG~II apparatus in the search for the charged-lepton flavor-violating decay
$\mu^+ \to e^+ a \gamma$, involving an axion-like particle in a $V\!-\!A$ chirality configuration.
While forbidden in the Standard Model, this decay is predicted in several Beyond Standard Model scenarios.
We performed Monte Carlo simulations with \textsc{Geant4} and analyzed data using the \textsc{ROOT} framework.
Our studies indicate that an optimized data taking configuration, using a low muon beam rate
($R_\mu \simeq 1\times 10^6\,\mu/\text{s}$) and a low photon energy threshold ($10~\text{MeV}$),
maximizes the experimental efficiency.
Analyzing only the 2021 and 2022 binded datasets, corresponding to aproximately on six days of data taking,
we show that MEG~II can reach a sensitivity limit of
$F_{V\!-\!A}^{95\%\rm CL} \geq 1.13 \times 10^9\,\text{GeV}$,
improving upon the 2015 TWIST result of
$F_{V\!-\!A}^{95\%\rm CL} \geq 1.00 \times 10^9\,\text{GeV}$.
This study demonstrates the capability of MEG~II to set world-leading constraints on this cLFV decay.