The diffuse supernova neutrino background (DSNB) is a powerful future tool to constrain core-collapse explosion mechanisms without observation of a nearby event, and the corresponding signal has been calculated for a variety of collapse models. For Supernova (SN) 1987A, a peculiar double neutrino burst was detected, but models for the double collapse have never been studied in the DSNB context. Here, we fill this gap and compare the DSNB signal expected in the Standard Collapse (SC) and the Double Collapse (DC) models in various future detectors, including Hyper-Kamiokande, JUNO, DUNE and the Large Baksan Neutrino Telescope (LBNT). We calculate the spectra of diffuse neutrinos and antineutrinos in the DC model and determine the rate of registered events as a function of energy of the detected particle, taking into account detector parameters. For each detector, we estimate the corresponding uncertainties and the background and compare the signals expected for the SC and DC models. We conclude that the combination of DUNE and LBNT data will have the highest sensitivity to discriminate between the SC and DC models.