Is the converter hard wired or does it have a cord you plug into a receptacle? If the latter, unplug it and go with battery for 12v and shore power for 120v. If ok, then plug something else in to that receptacle instead of the converter. Plug the converter into a different receptacle on a different circuit and see if that circuit now causes a GFCI trip. Use extension cords as required.
If the converter is hard wired, its black wire will go to a breaker and its white wire goes to the neutral buss. You can move the black wire over to a different breaker and see if the GFCI goes there now.
Is the converter chassis grounded to the frame of the RV or not? If so, try it with that chassis ground disconnected. 12v ground and 120v ground are shared in the Rv where both use the frame, so all sorts of funny things can happen.
I got GFCIs mostly from the outside receptacle getting water in it and it was on the same GFCI circuit as most of the inside receptacles.