My EVSE installation instructions said not to attach to a class A breaker ie a GFI type. I assume there might be a problem with two GFI circuits monitoring the same circuit. Don't know if this might increase interruptions with the S, though.
I've seen nuisance tripping with "series GFI's", when a GFI receptacle is connected to the load side of another GFI receptacle. That said, thousands of new hair dryers with GFI plugs are used with GFI receptacles every day. I know that I've heard the Roadster's UMC was pretty sensitive to GFI circuits. So there's an increased chance of a nuisance trip or a ground fault indicator. Technically, there should be nothing wrong with it.