As stated it can be done but it is costly.
Most are figuring you need so much power to run 8 hours a night 8 continuous hours. If running off genny all day and the trailer is already cool I would say using batteries 8 hours per night the AC would probably cycle on and off all night probably running at 20 minutes per hour on avg.
You could get away with 3 100AH Lithium batteries, a good charge controller and inverter. Solar panels would also help and can be had on the cheap around 100 a piece. The expensive part are the batteries, charge controller, inverter, etc.
Here is a link to a package for example https://battlebornbatteries.com/shop/chickery-travels-300ah-lithium-bundle/
You would also need to add a micro air easy start to the AC for it to ever start off of batteries. This kit with 4 solar panels, a 3000 watt genny to run the ac during the day and charge the batteries at the same time as the solar is helping will be in the $8000 range to have this all done and working properly.
Some people say holy **** that's a ton but if you really love to boondock and that's the primary way you want to camp and you have that investment for 10 years that's only $800 per year over that 10 years. Most spend alot more than that per year on full hook up campsites at moderate resorts per year. Heck you could spent that easily in 1 week per year at a nice resort campground.
It honestly is all in how you want to use your rig.