Ah, the adventure continues.
Enroute to the park we stopped for gas. While the pump struggled along, I unplugged the fridge outside, then had my wife turn on the inverter and attempt to activate the GFCI. No go. Shut off immediately.
We got to the park, got plugged in, and.....all is fine. GFCI hasn't shut off in a few hours, and we have full power. So, assuming it stays that way, the only thing that doesn't work correctly is when we're on inverter power. But, last night the GFCI shut off after several hours, so we'll have to see whether it stays on or not.
dave