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We investigate the experimental feasibility of detecting second-order double-magnetic dipole ($\gamma\gamma$-$M1M1$) decays from double isobaric analog states (DIAS), which have recently been found to be strongly correlated with the nuclear matrix elements of neutrinoless $\beta\beta$ decay. Using the nuclear shell model, we compute theoretical branching ratios for $\gamma\gamma$-$M1M1$ decays and compare them with other competing processes, such as single-$\gamma$ decay and proton emission, which represent the dominant decay channels. We also estimate the potential competition from internal conversion and internal pair creation, which can influence the decay dynamics. Additionally, we propose an experimental strategy based on using LaBr$_3$ scintillators to identify $\gamma\gamma$-$M1M1$ transitions from the DIAS amidst the background of the competing processes. Our approach emphasizes the challenges of isolating the rare $\gamma\gamma$-$M1M1$ decay and suggests ways to enhance the experimental detection sensitivity. Our simulations suggest that it may be possible to access experimentally $\gamma\gamma$-$M1M1$ decays from DIAS, shedding light on the neutrinoless $\beta\beta$ decay nuclear matrix elements., Comment: 8 pages, 4 figures, 5 tables