7-pin truck connector is for very low current, I think 10A max. For charging the battery it's usually OK, but not for running anything but trailer signals. Fridge in propane mode draws about 3A at 12V, so it still should be OK through 7-pin. I don't think your problem has anything to do with 7-pin or truck battery. If my memory doesn't fail me, that thin area between fridge and freezer has a door defroster wiring there or something like that. Open it and see if there is something wrong. Disable that defroster maybe.