I think it's gonna be hard to run your AC on the trail. Most rooftop AC's I'm familiar with are 110 and require being hooked up to a 110 outlet, which means shore power or a generator. You're not going to find shore power on the trail or in dispersed camping, but only in organized campgrounds (cost more for sites with outlets), RV parks, or someone's driveway. Running a generator will be a major nuisance to most other people out there within earshot. I know if I'm back country somewhere, I wouldn't want to hear a generator running all night. I'd probably be knocking on your door.
Don't expect to be able to run it from being plugged into an inverter that's hooked up to deep cycle batteries. You'd need quite a big bank of batteries to cover both the initial surge of the AC when it starts and then to keep it running for more than an hour or two.
Here's a link that explains it a bit better:
http://www.rv.net/forum/index.cfm/fuseaction/thread/tid/19812166/gotomsg/19812569.cfm
Here's another one:
https://www.offshoreonly.com/forums/general-q/82618-running-5000-btu-airconditioner-w-inverter.html
Other than that, I think it's a cool idea to try setting up a pop up for off-road. Though the axle may not be sturdy enough for dirt and gravel road bumping around, even if you don't do rough trails.
Don't mean to put pin in the balloon, but I think there are a couple things to overcome, main one being the AC, if you want your wife to be happy so she keeps going out there with you.