The tunnels will make you feel like you need to straddle the yellow line, but you only need to crowd it. Remember that the tunnel height posted is measured at the weight line. By the time you get to the center of the RV, it has already gained another foot or two.
Just the drive from Doughton Park (mm 241) down to Jullian Price (mm297) is a relaxing day if you take time to enjoy the sites.
If the campgrounds aren't open when you plan to go, there are alternatives. Black Balsam Rd at the Shining Rock Wilderness parking lot can be used for boondocking (
BRP Trip Report). Then just north of Mt Mitchell, take the forest service road across from Curtis Creek Rd, drop DOWN a couple miles to some road side marked campsites.
Try to plan for lunch or dinner at Mt Pisgah Inn, the food is great along with the view. When in the area, we usually plan a night at the Pisgah camprgound just to make sure we get a good meal.