Assuming that returning to a dealer for warranty repair is not feasible for some reason, you have two problems: the GFCI is tripping and the fridge won't run on propane.
First, the GFCI problem, you have two options - there is a ground fault in the trailer wiring somewhere or the GFCI is bad:
Make sure you don't have anything plugged into an AC outlet anywhere in the trailer. A prime location for a problem is the outside AC outlet. Disconnect from shore power and remove the cover and outlet. Check for insect nests, moisture, loose connections, etc. If you find anything out of the ordinary, correct it, put it all back together and test it.
There is another protected outlet in the fridge cavity. You can access it from the door on the outside of the trailer. Look in there and make sure that outlet isn't wet. Again with shorepower disconnected, remove the cover plate and outlet to check for insect nests, etc. After this check, leave the fridge unplugged from the outlet and try the shore power again. If it doesn't trip, there is a problem in the fridge. If it does trip, keep looking. The GFCI also protects the microwave. If you can access the power cord for it, unplug it and see if that corrects the problem.
If the exterior outlets look OK, you could disconnect again and check all the other outlets, but I'd be tempted to replace the GFCI breaker. Write down the part numbers and go to your local big box building supply and get a replacement. With shore power disconnected, remove the cover plate on your breaker box. There should be a screw on one side of the breaker with a black wire captured under it. Loosen the screw and pull out the wire. Then pull on that end of the breaker and it should pop off of the bussbar in the box. Snap the new one in and reconnect the wire. Replace the cover plate and test the system again. If it is still bad, you are in for a trip to your dealer.
Regarding your fridge - it should work in the absence of shore power, but it has to have 12V
. Do you have any lights on the fridge control panel? If not, it is not getting 12V
. Make sure your store/use switch is in use. When the breaker trips, do your lights work, and are they at normal brightness? If not you may have a battery problem. If you have lights, does the check light come on (if you have one on your model)? If so, it is not getting gas or is not igniting. You could have insect nests in the burner tube.
Sorry to be long-winded, but there is a lot to check if you can't go to the dealer.