I went through this issue on my fridge install. Inverter would run for a time, then the gfi on my 1000watt inverter would trip. I ran an extension cord from my inverter directly to the fridge and had the same issue. I solved my issue by replacing the gfi in the inverter with a standard outlet and it has been flawless since. This is one of those reasons why it isn't recommended to run fridges and freezers from household gfi devices. Minute current differences between the hot and neutral legs are what cause gfi trips.
2012 F350 6.7 L dually, 2013 3800RE with 6 pt leveling, Sumitomo 17.5" load range h tires, Samsung 18 cu ft residential fridge, 8k Morryde I.S. with disc brakes. Full timing since 2012.