First, I apologize for accidentally deleting Brownkevint's post that was quoted in the message above this one. Fat fingered mistake on my part.
Now, then ...
Was your tongue jack sitting on the ground at home, or was it on pavement? This would isolate the trailer frame from ground and keep the bathroom GFCI from tripping, even though the trailer frame was electrified. I assume the tongue or the stabilizer jacks were on the ground at the campground?
The problem is you have neutral and ground connected together in your teardrop. Since you're using a two wire cheater plug and the GFCI tripped at the campground, this confirms the ground in the box has a path to the trailer frame, then to earth.
This rules out a hot to frame short, which would give you a shock any time you touched the ground and the teardrop at the same time.
Of course, plugging in the two wire plug upside down, or plugging into a miswired outlet, will swap the hot and neutral lines, creating a hot to frame short. So it behooves you to find and eliminate the problem.
Like I said above, start by looking at your breaker box. Are the ground and neutral wires each on their own buss, or they sharing a common buss? Is the neutral buss insulated from the box via standoffs? If this is correct so far, is there a bonding screw connecting the neutral and ground busses together?
If you're sure neutral and ground are kept separate inside the box, and neutral is isolated from the box frame, then work outwards from there.