First unplug the trailer from shore power.
on the ground fault, with the power off see if u can go in and disconnect it. The GFCI has a supply side and a side that runs to the other outlets. u only need to pull the supply line (if u wind up changing it out, u want to mark the supply line and the line to other outlets). i'd put a piece of tape or wire nut over the bare wires, for safety sake, then plug back into shore power and see if u r still tripping the breaker. as koodog said, the GFCI could be bad and this would give u a quick test.
i'm thinking that the GFCI works by sensing a very small difference in the current flowing thru it in the neutral and hot. that usually trips the GFCI but not the breaker in the breaker box.
if it DIDN'T work with cord with no ground, i would tell u to unplug the converter then see if breaker trips. i would do the same with each item pluged into that circuit. BUT it work fine when on the cord with no ground...
if disconnecting that GFCI doesn't do it, i'm back to checking wires (screws) for tight in the fuse and breaker box.
although i've never seen a GFCI go bad like that, i believe it is a prime candate. (for the record, there are a lot of electrical problems i haven't seen).