The battery will run the fan and the "brains" of the furnace. Depending on how cold it is, i.e., how much the furnace needs to run, your battery could make it one night or a couple of nights. Remember, you don't want to run the battery down below 50% of charge; that will be bad for it. Most folks trying to do this either buy a small Honda/Yamaha type inverter generator or install a fairly large (45-100 watt) solar system to recharge the battery during the day. They usually invest in a true deep cycle battery, too. Usually two 6 volt golf cart batteries wired in series to give 12 volts.
PS: Your mattress heaters are 110 VAC, so they won't work at night (unless you run a generator).
1988 Coleman Sequoia - popup (1987-2007)
2012 Roo19 - hybid (2012-2015)
2016 Mini Lite 2503S - tt (2015 - ???)
2011 Traverse LT, 3.6L, FWD
2009 Silverado Ext Cab, 5.3L, 4x4, 3.73
2016 Silverado 2500HD Dbl Cab, 6.0L 4x4, 4.10