I've heard some places now require refrigerators to be on GFCI outlets in new home construction. Hadn't realized some RV manufacturers were going that way too. My 2016 Starcraft (division of Jayco) still has a regular outlet for the refrig.
Personally, I think it's a lousy idea for refrig or freezers. I don't think I've ever had a GFCI outlet at home that didn't eventually go bad. I've also had new ones trip with high humidity.
On my RV the AC outlet is just inside the vent door. Seems like a GFCI outlet would be subject to tripping during rain or high humidity in that case. Of course, they could use a GFCI circuit breaker, which is more expensive.
My ranting aside... I think I would try to isolate your problem. Maybe disconnect just the heating element and leave the rest of the refrig electronics connected? I imagine you've tried plugging other things into the outlet to see if it ever trips? Or replaced it?