We evaluate the cross section for diffractive bremsstrahlung of a single photon in the $pp \to pp \gamma$ reaction at high energies and at forward photon rapidities. Several differential distributions, such as the rapidity, the energy of the photon, and the azimuthal correlations between the outgoing particles are presented. We compare the results for our complete approach, based on QFT and the tensor-Pomeron model, with two versions of soft-photon approximations in which the radiative amplitudes contain only the leading terms. In the calculations we impose a cut on the relative energy loss of the protons, $0.02 < \xi_{1,2} < 0.1$, where measurements by the ATLAS Forward Proton detectors are possible. Our predictions can be verified by combined ATLAS-LHCf experiments. We also discuss the role of the $pp \to pp \pi^{0}$ background for single photon production. We discuss the possibility of measuring two-photon bremsstrahlung
in the $pp \to pp \gamma \gamma$ reaction.