To answer your question, simply NO. No your idea will not work. As others have already stated, you won't have enough amp hours for the required voltage to run the furnace.
Why don't you invest in a Honda Eu2000 generator? I use a single one all the time on "Eco mode" (aka: super quiet) at campgrounds. I use it to run the furnace all night long in the winter. Not once have I had a neighboring camper ever tell me that they could even hear the generator.
Then again, I don't camp anywhere where we're all packed like sardines in the camp ground. I've even run the darn thing where there's generator noise rules. One state park ranger drove by my camp at night. I flagged him down and asked him if it would bother him that I'm running my Honda "after hours" (on the quiet mode). He suddenly turned off his car engine and strained his ear out his window. He paused.............paused.......... Then said, "if I can't hear it, you can run it." He ended with, "have a nice night" and he drove away.
But seriously, that might work better for you. I usually camp in the winter time where the temps are down in the "teens." Two 12-v batteries didn't work so well with the heater running literally all night. Also, sometimes we have the heater off and 120v electric blankets running instead. The Honda on Eco-mode just purrrrs like a kitten all night.