I used to live in western Colorado and spent quite a bit of time adventuring around the Four Corners area. I would honestly recommend you NOT rent an RV unless you're already extensively experienced in driving one, because some of the most scenic and beautiful drives (even on paved roads) will take you up windy, narrow, drop-off-y mountain roads that will likely be very intimidating for someone driving an RV for the first time.
There's plenty of hotels/motels/bed and breakfasts in Utah, Colorado, and New Mexico (haven't been to AZ much but I would assume it's pretty similar), so you should be able to find something. Trust me, as someone who's done quite a bit of exploring around the Southwest, you'll be glad to have the SUV, even if you don't go on extreme 4wd roads.
Also, it's pretty easy to cook and eat paleo while camping, especially car camping. And it's fun! FWIW, I'm pretty sure Grok cooked his food over a fire, not on a mini-stove inside a heated/air conditioned RV. Get a cooler with some ice and restock it with meat and veggies along the way.
Also, if campfire cooking isn't totally your thing, you can pick up a small car camping stove fairly inexpensively to cook as well.
Hope this helps ... if you want some primal/paleo campfire food recipes or any advice on the best places to hit up, let me know, I'd be glad to share!