There's a rather elegant independent hotel on the estate. It's probably expensive, but you might want to check whether guests get discounted tickets.

There's also a winery with a huge tasting room, although obviously not for kids.

Would it perhaps be cheaper just to stay in one of the Biltmore hotels such as the famous Coral Gables property?

BTW, Ashville isn't that far from Spartenberg, where there's a BWM factory (assembly line tour) and museum. IIRC there's also a driving track where you can pay for lessons.