While it is not impossible to take an "A+toad" on the BRP, you will probably be much happier with the car-only experience. Some of the tunnels may require a change of underwear if you are driving the MH.
Tale a picnic each day and do one run North from Asheville toward Linnville Falls (stop at the Folk Art Center run by NPS) and another day South beyond Mule Pen.
Do Biltmore! Lunch at the stables.
In the Biltmore Village Shops there is a charming crafts gallery called New Morning. Some of the pottery and woodworking is outstanding. Several places to eat in the Village. We like the Station Restaurant.
Have fun