Unclear since you did not give the size of the battery you have. However, if you try and run the furnace
you are not going to get more than 2 days out of the battery and discharging the battery so much is going to severely limit its life.
Charging by running the car is going to be a bad idea. First you would have to figure out a system for connecting the battery to the car battery with some resistance, otherwise, you are going to kill your car battery by connecting battery directly to battery. Yes you do this when you need to because your battery has died, but it is not good for the battery to do repeatedly since the currents involved with connecting a dead battery to a live battery are very high.
Suggestions for dry camping
panels (30 W would make a huge difference with controller only cost me $100 and will save your battery)
2. Ditch using the internal water pump. I use a camelback resevoir in the cabinet above sink. Better pressure, uses much less water and no power required.
3. LED Lights
. better light
, won't kill the battery. The lights
that come with the camper will kill the battery quite quickly.
4. Turn the furnace
off while you sleep and bring a bigger sleeping bag. Quieter and won't kill your battery. Turn it back in the morning before you emerge from your cacoon.
5. basic math. Battery holds some number of amp hours which is stated on package. Everything requires some amount of current. Multiply usage time by current draw and compare to amp hours the battery can store. Battery should NOT be discharged more than 50% in any sort of regularity.
6. battery fullness is related to voltage and temperature. find a chart off the internet and a cheap multimeter or voltage meter to read battery voltage.