Thanks for the advice everyone. I do plan to keep it reasonably warm (65+) in the camper since my son is only 2 years old. I think I should be able to make it through at least one night, and I do have a battery meter, so if they're less than 50% the next morning I'll go for a drive or move to a parking lot to run the generator a few hours.
About 'stealing' power...I've been to less structured camping locations (horse shows, primarily...) where it's basically a free-for-all with extension cords and power outlets. Including plugging into someone else's generator, letting them plug into mine, or just charging my phone in someone else's trailer if they have power and I don't. I really wasn't sure what the etiquette on this was at an actual campground, which is why I asked.