Check your battery voltage. Get back to me. Probably the cause of the problem.
Say it is running 14.4 volts, that might be it.
Try disconnecting from electric power for a short time, and see what voltage you have. If still at say 12.7 volts (a full battery under some load) and it works fine, then you have a solution. Determine what circuit breaker feeds the battery charger, and you can turn it off when you get cold! At least this will work during this camping trip. Turn it back on a hour or so later, or when the battery gets low. While charging, the voltage probably will be low enough that the furnace might run.
Also check your battery water level. Your charger might be in a high amperage charge mode if the battery is low on water, or a defective battery charger might be overcharging it and boiled out most of the water. It is much better to use distilled water to refill the battery. However if the plates are exposed, and you leave the charger on at all (or start the RV engine) then the battery can be permanently damaged, so adding drinking water would be better than damaging the battery all the way!
Good luck!
Fred.