It will be easy enough to find out. You don't need to spend alot of money on a Volt meter. I would use this one Volt Minder. I would try to connect direct to your batteries. Once you have a battery meter at home set your furnace to come on about 5 or 10 degrees above outside temperature. Let it run all night and check to see what your battery level is in the morning.
VoltMinder Digital Volt Meter w/ 6 Foot Cord for 12V Batteries: Amazon.com: Industrial & Scientific
What I learned over several years of dry camping. When I had my single battery I could not even make it one night running the furnace set at 55 degrees inside at 40 degrees outside . We eventually stretched our dry camping into about 5 to 6 weeks without being plugged into shore power we had access to water and a dump station but no power. We use just a generator to charge the batteries. We beefed up our battery bank to four deep cycle batteries. I am not a power Mizer.. I watch TV with Satellite runing on inverters. I switched over half of my lighting to LED. I would run my generator (Honda 2000) about four hours a day. My battery charged would start out at about 13.2 volts and as time went on I would slowly see my battery level drop. After about two weeks...12.7...3 weeks 12.5 and by the 5th week 12.3. The story is four hours a day was just not enough time to FULLY charge the batteries. I have a 3 stage converter which charges at higher voltage at first then drops to a trickle as the batteries level starts to come up. Plust running a generator is not free. Even a small generator could drink a gallon of gas a day... A couple days not not much expense but over 30 days it could add up.
I have been researching solar. Problems with solar, COST although I have found a pretty good 300 watt panel with controler for under $700. Also I don't like parking in the SUN. I usally look for the best shady spot. Not too helpful for having solar. You could go with a portable panel but then the problems of hauling a large solar panel and theft to worry about. Eventually I probably will go with solar. As a Dry Camper it is the best option.