If you get outside of CA you might find some of those regulations ease up a bit. There are plenty of trails to take leashed dogs on but you must still pick up after your pet. Of course, no beaches here.
If you are going to leave them inside with AC and you don't have full hook-ups, you should go ahead and lock up your generator. That is no guarantee the generator will be there when you return.
Other options are to find wooded boondocking areas that provide lots of shade so the trailer doesn't get too hot in the first place and then use your fan off battery. Use your best judgement to not let the trailer heat up too much as trailers in the CA sun in warm areas can get pretty hot inside if you are not venting it with fans.