Try it with the inverter's chassis ground disconnected.
I had trouble with an inverter where its GFCI protected receptacles had their ground prong holes somehow connected to the inverter casing, which I had grounded to the RV frame.
The shore cord plugged into the inverter receptacle, has its ground prong wire going back to the RV power centre and is grounded there to the RV frame. It was all sort of insestuous wrt the neg side of everything.
With another inverter I had also with GFCI receptacles that didn't happen. Might be a Xantrex thing, no idea.