Agreed...weak battery. Another way to test it without the theory that the 120 V shore power is using some other ignition system, would be to hook up jumper cables from your tow vehicle to the battery in the trailer, and see if that gives you enough power to run the furnace.
I hooked up a digital harbor freight three dollar multi meter inside my tiny trailer with the sole purpose of determining at what voltage will my furnace stops working. Well I don't have the answer yet because I fell asleep when the voltage was down to 11.2 last time I remember looking. This experiment was well running off an old battery from a junk car.