We report on the investigation of dielectron production in tagged quasi-free neutron-proton collisions by using a deuteron beam of kinetic energy 1.25 GeV/u inpinging on a liquid hydrogen target. Our measurements with HADES confirm a significant excess of $e^+e^-$ pairs above the $pi^{0}$ mass in the exclusive channel $dp to npe^{+}e^{-}(p_{spect})$ as compared to the exclusive channel $ppe^{+}e^{-}$ measured in proton-proton collisions at the same energy. That excess points to different bremsstrahlung production mechanisms. Two models were evaluated for the role of the charged pion exchange between nucleons and double-$Delta$ excitation combined with intermediate $rho$-meson production. Differential cross sections as a function of the $e^+e^-$ invariant mass and of the angles of the virtual photon, proton and electrons provide valuable constraints and encourage further investigations on both experimental and theoretical side.