It would not be cheaper. At 6 to 9 mpg and 3,300 miles round trip with no detours, you'd be using 400-500 gallons of gas round trip, so say $1,500 in gas alone. You'd have rental on top of that and they're cheap, probably at least $2,000 per week. Add in night RV park fees during the AZ prime time probably $50-100 per night.
That said, I own a motorhome and would consider it if I had more time. It would be almost a 2-week trip minimum considering it would be a 3-day drive each way in a slow RV.
edit add: I see someone already responded.