Turn off breaker associated with that circuit, pull out GFI and direct wire the rest of the circuit without GFI. Flip breaker on. Does it stay on? if yes buy a new GFI and install, (they cost about 12$) if no, you have a short somewhere. I agree with replacing all original RV outlets. RV outlets are cheesy Chinese made CRAP. They fail and are notorious for overheating and causing fires. I would be suspicious of the outdoor outlets especially if they are crap Chinese made RV ones. They get water in them all the time causing GFI to trip. Replace them with an American made UL listed outlet.
RV crap outlets are really, really, cheesy and can be identified by no screw terminals onthe back of the outlet. Typically they press the wires into a U shaped lug inside the outlet right through the insulation and then snap a cover over the wires. DON"T USE THEM! If any large wattage appliance is plugged into them they overheat. I can't believe they are even allowed to use a non UL listed duplex outlets. They are really dangrerous and they will not pass a UL amperage test. They always fry.
-Paul R. Haller-