If your stay is primarily Biltmore-driven, Wilsons Waterfront RV park fits the bill, but there are no real amenities.
It is like an 8 minute ride to the Biltmore.
The park is clean, and level. There is a public trail that goes through the park that connects to a public park at one end, and another park along the French Broad river with playgrounds. Plenty of room to walk and bike. The waterfront sites are peaceful. Wilsons allows the use of Passport America quite liberally.
The only caveat is that it is right next to/under the interstate. If you can deal with the noise...you'll be fine.