I think you may be underestimating some of the distances you are looking at and/or overestimating how fast you'll be able to go, especially in an RV.

The Blue Ridge Parkway is a very slow going road. Its a fantastic scenic drive, but you're looking at maximum speeds of 45 mph, and may sections with even slower travel. It takes at least a few days to drive the whole thing, and you certainly won't be able to get from Charlottesville to Ashville in one day.

Nashville to Santa Fe in 2 days will be very difficult in an RV. It would be a speed run in a car, and RV's and speed run don't really work well together. I would plan to take at least a 2nd night on the road. Fort Smith AR and Tucumcari NM would probably make the most sense, with an easy drive into Santa Fe on your 3rd day.

If you can stay right at the campground within Grand Canyon NP, that would be your best bet, but you will need to make reservations.