Please use your multimeter to measure the voltage directly across the two battery terminals in these two conditions:
- Battery switch on
- Battery switch off
and report both voltages.
The furnace needs at least 10 volts or so to operate. I'm guessing that you are on shore power now. With the battery disconnected, the converter (charger) provides about 13 volts. If the battery is connected, and really low, it could pull the converter output below 10 volts.
If that is the case, and the battery is revivable, just leave the switch on for a while and let the battery charge before trying the furnace again. If you need the heat now, you can delay the charging until you don't need it.
Remember, though, that a discharged battery can freeze and be destroyed. A fully-charged battery can withstand a much lower temperature before freezing.