I'm picking up a Viking pop-up truck camper in the next couple of days. Need to get it ready for a trip in July to the mountains where I will not have shore power to plug into. I know a generator will...
The furnace is probably your only concern. If the starter batteries are going to be used to power the camper I would install a LVCO (low voltage cut off) leaving enough juice to start the truck in the morning. Personally I would add a separate battery for the camper, throw a cheap solar kit on it and not worry about using your starter battery as a deep cycle rv battery.