A number of theoretical calculations have studied the effect of radiation-reaction forces on radiation distributions in strong-field counterpropagating electron-beam-laser interactions, but could these effects-including quantum corrections-be observed in interactions with realistic bunches and focusing fields, as is hoped in a number of soon-to-be-proposed experiments? We present numerical calculations of the angularly resolved radiation spectrum from an electron bunch with parameters similar to those produced in laser-wakefield-acceleration experiments, interacting with an intense, ultrashort laser pulse. For our parameters, the effect of radiation damping on the angular distribution and energy distribution of photons is not easily discernible for a realistic moderate-emittance electron beam. However, experiments using such a counterpropagating beam-laser geometry should be able to measure these effects using current laser systems through measurement of the electron-beam properties. In addition, the brilliance of this source is very high, with peak spectral brilliance exceeding 10(29) photons s(-1) mm(-2) mrad(-2) (0.1% bandwidth)(-1) with an approximately 2% conversion efficiency and with a peak energy of 10 MeV.