From what I can gather a furnace blower typically pulls about 7-10 amps, depending on size. I'm picking 8 amps without knowing the model of the furnace (RV is in IL right now and I am in TN.) So if the furnace runs say 8 hours out of 24 (very generous estimate), that would be 64 amp-hours.
Our other battery usage is very low---no big inverter for microwave or coffeepot or hairdryer. We run the water pump and LED lighting. We have a 12V outlet that we use with a 400W inverter to charge electronics, but that is often done in the tow vehicle while out and about. Very little or no TV use.
I'm going to estimate 80 amp-hours/day and I think that's generous based on our habits.
So we need a converter that will replace that 80 amp-hours in a reasonable amount of time, say 4 hours or less. That means a converter with a bulk charge mode of 20 amps or higher. So a 30 or 40 amp converter should serve us well.
We will likely be at an electric site at least once a week, for 24 hours or longer. So if we get the batteries up even close to 100% otherwise, during that time frame they should definitely get maxed out and be just fine.
Does my reasoning make sense? Any big flaws in my logic?