We report on a study of the Single Event Sensitivity (SES) 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 (ALP) in a V-A chirality configuration. While forbidden in the Standard Model (SM), this decay is predicted by several Beyond SM Physics scenarios. We performed Monte Carlo simulations with GEANT4 and analyzed data using the ROOT-Cern framework. Our results indicate that an optimized configuration, using a low muon beam rate ($R_\mu \simeq 1\times 10^6\,\mu/\text{s}$) and a low photon energy threshold (10~MeV), maximizes experimental efficiency and the SES. With only a few days of data taking, MEG~II can reach a sensitivity limit of $F_{V-A} \geq 2.3 \times 10^9\,\text{GeV}$ at \,95\%\ \text{CL}, improving upon the 2015 TWIST result of $F_{V-A} \geq 1.0 \times 10^9\,\text{GeV}$ at 95\% CL. This study demonstrates the capability of MEG~II to set world-leading constraints on this cLFV decay.