The furnace burns propane to create heat, but the control board, ignition and blower are all ran from the 12 volt DC power. Have you been having 12v issues? Why did you add a battery? Are you able to use a voltmeter to test you batteries and power supply? What type of shore power are you getting, a full 30amp RV service connection?
I would start with checking battery voltage and charging. Disconnect from shore power. After about ten minutes, check the voltage at the batteries, individually and together in parallel. What are those numbers? Plug back into shore power. Test the batteries again. What are those numbers? They should be higher, showing the batteries are charging.
You could also have a malfunctioning thermostat causing the furnace not to turn on below 50 degrees. They are known to go bad and are often easy to replace.