I missed the a/c part. Campers wired for a/c typically have a complete 30A@120VAC package installed consisting of a heavy shore power cable with a 30A@120VAC plug and a set of 15A@120VAC circuit breakers in a box or the converter.
One of the breakers will be dedicated to nothing but the a/c (and its heat strips, if any) and the other breaker will be for all the rest of the appliances, converter and outlets.
If someone has a 15A@120VAC plug on the shore power cable and also has a/c installed, they have a jack-leg, non-Code setup and should consult a knowledgeable electrician.
Not sure I agree with the 'different prongs at different times' explanation for a GFCI tripping. The way it works is that the GFCI is balanced and more current on one power leg than on the neutral leg implies current flowing elsewhere and the unbalance opens the circuit. If one prong is touching and the other isn't, there's no current flow unless there is leakage to ground...
WiKi on GFCI
1. I don't believe all GFCIs will detect a missing ground, just an imbalance, however it's likely the test button wouldn't work.
2. I don't believe all GFCI will detect reversed power and neutral, again it needs an imbalance to trip. There are different types of GFCI, some may detect the power wire, some may not.
3. I suspect any GFCI breaker that is doing strange stuf is either strange itself or is a strange situation...
4. Contrary to popular understanding, not all GFCIs protect outlets 'downstream' from them. I have personally handled a GFCI receptacle which had wiring options to protect or NOT protect the downstream outlets.