If a GFIC is tripping, you HAVE a ground fault.
Since it is NOT happening when the RV is plugged into shore power, but happens when you are running on the genset, I will suggest you have a neutral-ground bonding issue in the transfer panel.
Safely grounding a 120/240V electrical system requires exactly one (no more, no less) point of bonding between the ground lug (round, centered hole) and the neutral (return) leg (one of the slits).
When plugged into shore power, the bonding point is inside the MAIN breaker panel.
When running on an independent genset, the genset or it's distribution panel must provide the bonding.
As an aside, backup generators for a house use the main panel's bonding.
So, the transfer switch has to provide ground-neutral bonding when running on the built-in genset. Or, if you are using a portable genset, you have to add it when running the RV and remove it when powering the house during a power failure.
TT 2015 19RD "TheJayco"
TV 2003 F-350 "Montblanc" - Housebroken chore truck
Sitting in The Cheap Seats.
And proud of it!