OK, a couple of things.
The Coleman Gas Lantern should not be used inside the camper. Not only is filling it dangerous, (I have some pictures of a travel trailer burned to the frame due to fueling a coleman latern inside the camper) but the carbon monoxide is odorless and poisonous. The camper is much better sealed than a tent or popup. Opening a window will vent more heat than the lantern produces in any case.
Another issue is the battery. Not sure what the amp hour capacity is on your unit (the one shipped with my camper is a Deka DC-24 rated at 75 amp hours), but there are several very good discussions here about battery capacity and usage.
Running the furnace is one of the biggest 12 volt draws in the camper.
The blower will deplete the cheap deep discharge battery they install pretty darn quick.
It can take a day or two, not hours, to replace that charge.
I think you will be running that generator a lot longer than you think, so
bring lots of gas for it.
Lou and Laura with Bella - German Short Hair Pointer
2008 GMC Sierra 2500HD Crewcab SB Allison Duramax
2010 Flagstaff 8526RLWS - Superglide 3300
HAM CALLSIGN - KC3FFW