The electric heating elements can trip the GFCI. You should try replacing it. I had one that occasionally would trip the GFCI when I first turned the refrigerator on. It would automatically switch to gas at that point and go ahead and cool down. When I'd notice the gas was doing the cooling, I reset the GFCI and it would work fine after that making it hard to find a short in the element with a multimeter. The heating elements aren't that expensive. Just go ahead and change it and see if it fixes your problem.