Good grief..
You do operate your fridge on gas? Correct?
It works, right?
Furnace is POWER VENTED, in other words the exhaust is forced by the same fan motor as the interior fan..
Furnace flame should also still stay on just like the fridge gas burner while traveling. If it does go out, furnace board will attempt to relight and continue just like your fridge..
Generator carbs tend to dislike the movement that it may encounter when driving and most likely will stall out..
Try running the built in furnace before traveling down the bunny hole with heating with electric heaters and a generator..
OR, better yet, just leave your water system winterized until you reach your destination.. Then rewinterized before leaving for home..Everything else in the trailer will most likely survive the cold temps on both ways..