I know that due to the huge mass difference and dependence of radiated energy on mass, protons lose much less energy in synchrotrons than electrons.
Can someone tell me how to calculate the energy limits at which the energy radiated by the protons in synchrotron cooling becomes significant?
A precise calculation require additional information on the machine layout and optics in addition to the mere particle energy (see: http://pcwww.liv.ac.uk/~awolski/Teaching/USPAS/FortCollins/USPAS2013-DESDR-Lecture1.pdf)
Anyway consider that the emitted power $\displaystyle\propto \frac{E^4}{M^4 R}$ where $E$ and $M$ are respectively the kinetic energy and the rest mass of the particle being considered, and $R$ is the bending radius of the dipole magnets in the ring.
Since $M_p \approx 2000\,M_e$, if you would like to put protons in an existing electron ring and get the same amount of radiation, you will need to crank up the energy by a factor $2000$. You will be impinged by the maximum magnetic field achievable a long long way before hitting that energy. If you go for a larger ring radius then you will need to increase the energy even more.
For reference at the LHC, with $E=6.5~$TeV and $R=2800~$m you get a damping time of the order of ten hours, however the cooling effect is partially shadowed by intrabeam scattering (IBS) and other non-linear effects. Still the emitted radiation is enough to allow for a telescope (BSRT) to capture it giving a bunch-by-bunch transverse image of the beam.
That being said, synchrotron radiation is not the only mean to cool a beam. Stochastic cooling, electron cooling, laser cooling, ionization cooling (more effective with muons) are all possible viable methods to reduce the emittance of a proton beam.
