An interesting problem.......and my ramblings are attached. If I understand your situation correctly: your “on board” RV generator starts and after the short time delay the transfer switch operates to cutover to gen power. Is this when the GFCI trips? Once fully cutover and running on gen power you can reset the GFCI and it functions and powers its loads normally and can be manually tripped as well? Nothing else has changed in your coach AC electrical system that would explain the GFCI trip; AC loads on the GFCI, moisture, wiring changes etc. ?
My understanding is that the RV AC panel must not have ground and neutral bonded together. The requirement for a single point of ground and neutral bonding comes from the shore power source (when connected to shore power). I suspect that most “on board” RV gen sets have their neutral and ground wires bonded at the generator somewhere, so that when the transfer switch cuts over to the generator, that single point of ground and neutral bond still applies (supplied by the generator).
So if your GFCI trips during the initial transfer switch operation, but can be reset and operates normally after the gen is powering the coach AC loads, then something would seem not correct during the actual transfer operation......like maybe in the transfer switch the hot contact makes before the neutral contact makes to the generator feed. This might look to the GFCI like current is flowing from hot to ground for an instant when transfer switch contacts are in transition. Once both hot and neutral contacts are fully closed to the gen feed, then normal hot and neutral current flows allowing the GFCI to be reset and function normally. So is it possible for the transfer switch contacts (hot and neutral) to no longer make at essentially the same time? Not sure.....perhaps if the hot contact has a build up of contact deposits and the neutral is burned or pitted down from long use under high load? A head scratcher.