The furnace is what will deplete your battery. Not running the furnace, using LED lights or being very careful with how much lighting you use, you could go for a few days on a single "good" battery.
I would recommend a second battery, charging it while you drive, and swapping them even it is a bit of effort.
The reasons are;
paralleling a charged and discharged battery means you have two discharged batteries. It is not efficient.
inverters are not power efficient. You would have losses going from 12 volts to 120 volts, and then back to 12 volts. The end result is you don't benefit from the fully charged battery.
Now having said this, how close do you park the truck to the pop up? Would it be close enough to connect a cable from the battery you have charged in the truck to the pop up? Heavy welding cable with crimped connectors would work for this. In fact, if your battery is good, this may save you from buying a new battery. Not much different then charging the pup while you are driving.
