It's true heavy furnace use will kill a battery pretty flat. Depending on your converter size, it will charge up in a few hours of fun time.
If you do a lot of dry camping you'll likely end up converting to a multiple battery setup, dual 12v or 6v, ending up with 200ah or more.
I dry camped last fall, in sub 30* temps. I've got four 6v batteries in a split parallel arrangement giving me a total of 460ah. Furnace set to 50* and running my cpap, I would be down about 70-75% or 12.4v. It would take about two hours for my converter to reach float mode, on one Honda EU2000i.
You need two 2000w units to run your AC, but one will run everything else.
Now-2014 Sierra 346RETS 5er BUB
Then-2002 Keystone Springdale 286RLDS TT
Nights camped in 2014-28, 2015-127, 2016-10