assuming knowledge of the geometry of the cavity (i.e. its length) and the free spectral range of the modes, as well as the frequency of the light that it emits, Is there a quick and easy way to calculate the FWHM of the gain profile?
First, because normally, gain profile refers to a property of the gain medium only, independent of the cavity geometry. So the parameters you specified have nothing to do with the gain profile.
The gain profile will depend on the composition of the gain medium and on the pump power applied. Normally the gain profile is much wider than the spectral width of the resonant modes of the cavity. Because of this, the "FWHM" of the gain profile isn't something that's even normally calculated.
Second, if you actually meant the spectral width of the output beam, you can't easily calculate that either, because it depends not only on the cavity length, but also on the mirror reflectivities, optical loss in the cavity, and phase noise introduced by spontaneous emission in the gain medium. It may also depend on engineering details such as "mechanical vibrations, temperature fluctuations, and pump power fluctuations" (source).