Soon available multi-petawatt ultra-high-intensity (UHI) lasers will allow us to probe high-amplitude electromagnetic fields interacting with either ultra-relativistic electron beams or hot plasmas in the moderately quantum regime. The correct modeling of the back-reaction of high-energy photon emission on the radiating electron dynamics, henceforth referred to as radiation reaction (RR), is a key point for UHI physics. This will lead to both validation of theoretical predictions on the photon spectrum emitted during the laser-particle interaction and to the generation of high-energy photon sources. In this paper we analyze in detail such emission using recently developed models to account for RR. We show how the predictions on the spectrum can be linked to a reduced description of the electron distribution function in terms of the first energy moments. The temporal evolution of the spectrum is discussed, as well as the parameters for which quantum effects induce hardening of the spectrum.