When dense high-energy lepton bunches collide, the beam particles can experience a rest-frame field which greatly exceeds the QED critical one. Here it is demonstrated that beamstrahlung efficiently converts lepton energy to high-energy photons in this so-called supercritical QED regime, as the single-photon emission spectrum exhibits a pronounced peak close to the initial lepton energy. It is also shown that the observation of this high-energy peak in the photon spectrum requires to mitigate multiple photon emissions during the interaction. Otherwise\comm{,} the photon recoil induces strong correlations between subsequent emissions which soften the photon spectrum and suppress the peak. The high-energy peak in the photon spectrum constitutes a unique observable of photon emission in the supercritical QED regime and paves the way to a laserless gamma-gamma collider based on electron-electron collisions. Show Partial Abstract