On my old Rig with a Freedom 25 inverter, the 110v power travels first to the Circuit Breaker Panel.
Then for the selected Inverter controlled circuits the 110v wiring goes to the Inverter.
From the Inverter, the 110v power goes to the GFI outlet (I only have one GFI located on the same wall as the CB Panel).
Since you have one GFI working and one GFI not working I would think you have 110v to the Inverter, but only one leg outbound to the working GFI outlet.
So the problem might be a weak GFI (which you have tested) or a weak breaker or circuit on the Inverter controlling that non-working GFI?
If you replace the problem GFI outlet with a standard 15a outlet, do you still have the same symptoms?