Have you measured the voltage at the battery? At the battery while under load?
Most furnaces will ignite at 11.6 volts, and that is a battery that is about 20% full. Your engine alternator is probably putting out about 14.5 volts and will allow the furnace to run while trying to charge the battery. Once the engine is off, typically a 'full' battery will have between 12.7 and 12.8 volts, while one just off the charger might have as much as 13.3 volts for a 1/2 hour, then will settle to 12.8 volts.
Your RV will use 1.2 amps per hour just to run the CO detector, propane monitor and refrigerator. This is enough to discharge a full battery in about 2 days, even with no lights running. Add a 6 amp per hour furnace, and the battery can go dead in a few hours.
Are you plugged into shore power? If you are, then the converter / charger should be supplying the power to keep the battery full. Check for a tripped GFI someplace that might be supplying power to the converter. If there is no power to the receptacle it is plugged into, try running another extension cord from a good power source to the charger. Even if you do not have a digital volt meter, you can look at your gauges and see if battery reads 1/3 or full. IF it reads empty, and you are on shore power, suspect a problem with your converter.
You have a 1984, probably is a converter, not a modern battery charger. You would be much better with a modern charger, but they are around $300 - $400. They can put out up to about 45 amps, and accurately charge the battery. However the prior owners might have already upgraded to a charger. You will need to look for it, and report it's model number if you have low battery voltage and problems.
Good luck,
Fred.
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