We present preliminary results of our study on the impact of theoretical uncertainties on distinguishing the Standard Model (SM) two-neutrino double beta decay ($2\nu\beta\beta$) from beyond-the-Standard-Model (BSM) contributions. To this end, we employ a frequentist statistical test and focus on distortions in the single and summed electron energy spectra, as well as the angular correlations of the emitted electrons, with particular emphasis on theoretical uncertainties arising from phase-space factor (PSF) calculations and nuclear modelling. As a BSM process, we consider $0\nu\beta\beta\phi$, neutrinoless double beta decay ($0\nu\beta\beta$) accompanied by the emission of a light scalar $\phi$ -- often referred to as Majoron in the context of Majoron models and the spontaneous lepton number violation -- and we derive limits on the effective scalar-neutrino coupling.