Shooter's Choice / Hoppes' No. 9 for general cleaning, Eezox for a 'finishing touch', CLP for moving parts. Nowhere is it written that it's got to be just one product or another; IMHO combining a few different ones works much better as different cleaners are slightly better at different things.
Treat a stainless gun as if it were a blued gun, and you will not ever have a problem.