The emission power of synchrotron radiation depends on the relativistic Lorentz factor γ to the fourth power, so 10 GeV electrons radiate much more than 10 GeV protons, other factors being the same. See
Protons do radiate synchrotron radiation. It is expected to be a serious problem in LHC at full energy (7 GeV per beam), because the proton synchrotron radiation is soft x-rays, that cause residual gas atoms adsorbed to the beam tube walls to desorb, causing a vacuum pressure rise and proton residual gas scattering and beam loss.