I think it's campground specific. I talked to them and they said it was fine to run a generator and she didn't list restrictions. I am just super conscious about noise because I can't stand other people running their generators for extended run times. We will be in the mostly tent section so it's even more important to be thoughtful of others.
If I run the generator, I'm only thinking of running it in the evening to cool down camper when babies are ready to go to sleep. The issue is the latent heat buildup in the materials of the camper, it takes a while for that heat to come out so running a generator isn't just a matter of a few minutes, it's about an hour or more.
At this time, the two week weather forecast has weather looking decent enough that we might not need to worry about the heat for 5 of the 7 days. Thankfully it is cooling down at night into the 60s and that will be a huge help. The last two nights are trending hotter but we can either head home early or run the onboard generator to get us by till the night air is cool enough to manage.
If it was a week of hot weather like I thought when I first posted I'd probably still buy a gas generator but with two nights now I don't think I can justify the cost.