If you can swing the finances, it could well be cheaper to buy a used RV and then sell it afterwards. Of course, you do also run the risk of needing some expensive repair that both throws off your schedule and costs as much as any savings—which can to some extent be prevented by buying carefully and checking thoroughly before you go.
Driving a class C is not all that difficult, though it certainly is different than driving a car and requires somewhat of a different mindset at times. I find it's kind of like driving a U-Haul, except that my class C is more comfortable and has better visibility because of windows in the house section. Length makes relatively little difference for how hard it is to drive on the highway (though a longer unit is more cumbersome in parking lots and gas stations and similar tight quarters). You do have to give yourself more space to brake and accelerate and anticipate possible difficulties somewhat further ahead, as stands to reason with any fairly large and heavy vehicle.
The itinerary sounds doable to me. You're averaging somewhere in the vicinity of three hours of travel time per day over a five week period. Obviously, some days will be longer and some shorter, but that's not at all an unreasonable pace in my opinion.
I would very much suggest that you try a quick trip with an RV before setting out and probably before finally deciding to set out. While travel by motorhome is wonderful, you might find that it's more comfortable for you to tent camp and/or stay in motels rather than deal with driving a big RV or handling all the various systems. It would also help you see how well your dog adapts to RV travel, and maybe gauge how likely it is to bark incessantly if you leave it alone in the RV.
If it were me, I would very much consider leaving the dog behind for the trip. Many places (not so much campgrounds, but attractions) are not especially welcoming of pets, and so it would probably spend a lot of time cooped up in the RV.