I have not stayed there, but
Lake Powhatan Campground and recreation area is probably the closest one you will get. It is accessed via Bent Creek Branch Rd/Wesley Branch Rd, which runs off of Brevard Rd/NC191 about 2000 ft from the Blue Ridge Pkwy interchange. Google says 14 miles via I-40 or take the BRP across to US25 and do the loop to get to the ticketing area, 15 miles.
Also visit
The North Carolina Arboretum whose entrance is even closer to the BRP interchange. $16 parking fee, free to enter, some hiking trails but a wonderful place to spend an afternoon.
If you will be traveling further south on the BRP, know your MH height, as the tunnels get more frequent and lower the further south you go.
You can camp on the Blue Ridge Parkway in several places.
Julian Price Campground, is close to Blowing Rock.
Further south is
Linville Falls Campground and the falls themselves, with hiking trails. The movie Last of the Mohicans was filmed at various places in NC and some of the scenes were shot at Linville Falls. There are no hookups, just a dump station and scattered water hydrants.
The
Museum Of North Carolina Minerals is located at the BRP interchange with NC225 about 3 miles before Little Switzerland.
Crabtree Falls Campground is about 6 miles south of Little Switzerland and has a very nice water fall that is about a mile hike, but its a workout.
15 miles south of the Brevard highway interchange on the BRP is
Mount Pisgah campground, which is my favorite at 4900 ft elevation.
This is leaf peeping time on the BRP and the whole area so expect traffic and congestion wherever you go.
Charles