At a quick look, it seems FAIR1.3 already has some stratospheric components (ozone, water vapour) so it's probably not too difficult to extend that to sulphur aerosols with some simple approximations for aerosol lifetime and proxy interaction with ozone. Someone else better versed in physical climate modelling for SAI would have a more complete list of dynamics that'd need to be represented though, but it may not be too important given that it is a fairly simple model of emissions and radiative forcing.
I put together an updated module for SAI in the RICE50+ model (itself an update of the RICE/DICE family), although that was a much more simple implementation since the climate module there is basically just MAGICC6, so it was ultimately just adding an extra term to the forcing calculation. I imagine it'd be somewhat similar with FAIR1.3, except you'd want to actually model the stratospheric aerosols themselves.
Given the discussions from the CCIS workshop last year on how we should look towards using emulators more, FAIR is probably a good starting point.