Hyperons and their electromagnetic decays have become one of the major objects of interest of the HADES experiment, providing access to the inner structure those hadrons at the femtometer scale. With a recently collected data sample of reactions produced by a $4.5\,\mathrm{GeV}$ proton beam impinging on a liquid hydrogen target, a new avenue of such studies has opened up. Several steps are necessary in data pre-processing, such as quality assurance, calibration and normalization, before any data analysis can be performed. A "standard" reaction, namely elastic $pp$ scattering, has been proven to be a versatile tool that can be applied to tackle those. A feasibility study for a planned measurement of the "unknown" Dalitz decay of the $\Sigma^0$ hyperon, produced in $pp\rightarrow pK^+\Sigma^0$ reactions, is performed. This would be the first measurement of an electromagnetic Dalitz decay of a hyperon. Two complementary analysis strategies are developed for this purpose. Preliminary simulation studies show that a pioneering measurement of the $\Sigma^0\rightarrow\Lambda e^+ e^-$ decay is feasible.