We've left our gen. in our TC running with the AC a few times for the dog. It was normally for only a few hours while we visited some sort of attraction. I've never had any problem with it.
However, unless it's really really hot I don't think the AC is necessary. We would often just leave the windows open and the Fantastic fan running. This alone kept it cool enough.
If we're just stopping for a short time like to eat at a restaurant I found it's easier to just leave the truck running for the dog. This saves moving the dog to the TC and then moving her back to the truck. It also keeps the truck cab nice and cool for us. It also doesn't use up my propane supply in the TC.
For your application of a rented class C I think leaving the gen. running should be fine. Just make sure you have a good bit more than a 1/4 tank of gas.
