My guess is you have not added the proper amount of distilled water to the battery over the years, and now the battery is somewhat dry, and will require replacement.
Take off the battery caps. If you can see the lead plates inside, not covered with water, then the batteries are probably bad. If you see the battery water is full, then the water will be up to the little plastic things that reach about 3/4" down below the top of the battery, with a 1/4" deep air gap once the battery water is "Full" as it normally should be.
If the battery is low, then add only distilled water to it, to bring it up to the proper level (not the top).
Then shut off the "Conv" circuit breaker. This should shut off the fan, and stop charging the battery. IF the lights suddenly dim, you probably have a bad battery. If you have a volt meter, check the voltage. If it reads 13.2 - 14.0 volts while the "Conv" converter circuit breaker is on, then that is normal. If this drops to about 12.8 volts after shutting off the converter for a minute or so, that is a normal voltage for a battery in good shape. If the battery drops below 12.6 volts within 10 minutes, it would indicate a bad battery, that will not have the strength to last more than about 4 hours on a night when you need the furnace.
If the battery voltage drops below 10 volts, that indicates that there is a shorted cell within the battery. You must replace that battery.
Chances are that you might have a shorted battery, and as soon as you plug into shore power, the converter goes to 100% output, then the fan comes on. IF the battery was good, then the converter might see 12.5 volts on the good battery, and not go into 100% output when you plug in and not turn on the fan. ALso the good battery might only accept 20 - 30 amps - thus the fan will not need to run.
With a bad battery, you will normally hear the fan all the time.
Good luck!
Fred.
PS: I also think that you have the terms in-correct. Most every RV will have a "Converter" or since 1992 a electronic battery charger that is sometimes called a converter or converter/charger, or sometimes just a charger.
Back in the 70's and 80's the RV's did have "Converters" that converted 120 volts AC to 12 VDC. They where not well regulated, so a campground with 100 volts input would mean dim lights and only about 10.5 volts output, so the battery will rarely charge at all with such low input voltage. At 120 volts input then the converter will put out more than 13.5 volts, boil away the water, and likely also cause the light bulbs to overheat and burn out quickly!
In the 80's and 90's, inverters became available at a much lower cost. Prior to 1983, a 50 watt inverter that could only run a telescope timer motor might cost something like $250. By 1990 you could buy a 150 watt modified sine wave inverter for about $150. Now you can hardly find such a small inverter, and the cost rarely is over $50, even for a 500 watt model.
Sine wave inverters are still more expensive than the modified sine wave inverter. All inverters take 12 VDC (or another battery voltage) and converts that to normal household voltage. (120 VAC 60 Hz in the United States, and 220 volts 50Hz in other parts of the world, such as Europe.
Fred.