It is hard to overstate how much power a furnace can use, if it has a fan. A blind estimate of your trailer would be 10 amps. We use our travel trailer in the summer, and can go a week or more running LED lights, TV, fridge, and so on. Go out in fall when it is below zero, and the batteries last 2 nights max, and I have (2) 6V golf cart batteries, which store more than your standard size batteries. They weigh 80 lbs each.
I would assume that your furnace runs a lot if you are in there day and night, and as has been noted above, your charger is not able to recharge the battery in a reasonable time. A Honda 2200 generator can produce 2000 W, +/-, but your charger is only producing 3.3A*12V, or about 40W. If you go with my guess that your furnace fan uses 10 amps, you need to recharge 3 hours for every hour of furnace time.
You need a faster charger, and probably even one that can charge both batteries separately if you are going to get them charged in a reasonable time. I think you can safely charge one battery at 15A, so if you charged both separately, you could charge 10X faster than what you have now.
All this is guesswork, and like others have stated, a meter is what you need to find out what is really happening. There are meters you can buy for not much that clamp over the wire and measure the current in the wire. That will let you know how much you are using, and then you can calculate how much battery you need if you want to last a few days without recharge.
Taking batteries to zero volts, especially in cold temperatures, reduces their life a lot, and you will be replacing them frequently if they are left dead out there in winter.