I live in Oregon and travel/camp all over the west. I don't believe I've ever made a campground reservation. Aside from showing up at canyonlands national park and secure three sites on the white rim trail anyway. Then again I try to stay away from the huge touristy campgrounds with swimming pools and such.
I don't know where you're going on the north rim but toroweap campground is amazing if your rig can stand a bunch of washboard roads.
The southwest is going to be extremely hot in June too.
I don't believe they're suggesting idling the engine overnight.
Between having a large battery, solar, energy efficient components, short drives should recharge the battery and a short time idling. Their setups really shouldn't need to have constant charging power.
When you start driving off in an earthroamer you ought to be working it enough to clean it out.