Im thinking a ground problem, or weak GFI. Here are a few ideas:
*Get a multimeter! Preferrably a DC Amp clamp meter. (you can get one on amazon for about $50) MOST <$100 clamp meters do NOT measure DC amps. So read the features carefully.
*Try turning the converter off and see if the problem happens. If the problem does not happen, verify that ground and neutral are not tied together on the converter.
*Multiple GFI's in a chain can be interesting if one is going bad. ONE in the chain can trip them all. Even if that one is not being used. There is a GFI check tool but I am unsure that will help because it seems the GFI -is- working, its just too sensitive. (only as strong as the weakest link)
*Is there anyway to isolate one branch at a time and have someone help you watch the plugs to see which pops first? (this may not be possible at all) A little more work would be to disable the plugs one at a time and see if the problem stops.
*Because this is only happening on shore power,I'm leaning toward the shore power ground is 'better' than the ground on your camper so the GFI is having a fit. A fix for this is the dreaded grounding rod, and grounding your camper directly to Earth and seeing if that fixes your issue. OR.. attach a ground wire to the water spout at your camp site (assuming its metal)
*I'm not surprised that it all works on your generator, which is about the same quality ground as your camper, which is to say your batteries.
*The multimeter should have continuity check ability (it beeps when a direct connection is made).
Check your cable to verify the + is only touching + and the - to - etc. Shake the cord and bend it a bit to make sure there is no 'accidental' connections esp. to ground.
Hope this can help!
GFI in my house kept tripping whenever anything was used. I was stumped. It was another plug in the chain that was faulting. Replaced it and no troubles since.