What ISO speed would you recommend for doing photometry during a full moon or even if you change the ISO speed based on sky brightness at all?
It's impossible to recommend a specific ISO value without knowing more details about your setup: And even then, ISO numbers are not what used to be. They had a very specific meaning back in the time of chemical film, but nowadays with digital sensors, while camera vendors might try to maintain some loose level of continuity, I would not rely on it being comparable between vendors or models.
The effect of sky brightness on photometry depends on the focal length, but anyway, if you feel like you have to change the ISO gain in moonlit nights to avoid saturation of star images at your typical exposure time, you might already be (too) close to saturation even in darker nights as the background should ideally contribute only a small fraction of your measured photons. You don't want to operate too closely to saturation levels anyway. If the sky background really has a huge effect on your measurements, can you move to a longer focal length perhaps ?
Sensors (or more precisely their readout logic) can change their characteristics (e.g. noise level) rather dramatically from one ISO (=gain) setting to the next so as a general guideline I'd advocate to
a) assume nothing, instead
b) test different settings and compare results for consistency and quality, but then
c) stick to things that work well and keep it simple by not jumping back and forth unnecessarily between different settings
I guess without more concrete infos on your setup, general rules of thumb are all I have to offer, sorry :-)