There is no GFCI on refrigerator circuit in the camper, so I assume the issue is GFCI on the house, where the cord is plug in?
If the camper is powered fine with refrigerator off and trips GFCI when you start the fridge, it is clear indicator that the fridge has small AC leak and GFCI is protecting you from being "link in AC leak".
I would pull the panel on the back and plug the fridge directly to the cord to confirm, but full testing would be a case for good electrician.
Still what you can do is check for wires on fridge rubbing against the frame and other suspicious elements.
Garage circuits in CA should be GFCI as well. No harm plugging into 20 or 15 amps as camper has its own breakers.
