The only horse camp that I know of in the area is Blowing Rock Equestrian Preserve -- but I suspect that's the place that you already know about.  I will say that their stable facilities are okay, but as you mentioned, most camping is done "in a parking lot".  The only exception to that is if you can get a spot in that uppermost camping area (on the left, before you reach the arena), that one is a little more shaded and a little nicer than the others.  But BREP is not a place that I would go for the campground amentities, the only thing it really has going for it is the direct access to Moses Cone trails!  But man, oh, man the trails at Moses Cone sure are wonderful!

Callalantee Stables used to be located in Mountain City, TN which was about and hour's drive from Blowing Rock.  I think I heard that they turned that into an ATV place, not for horses any more but I don't know for sure.  Also, I heard about Leatherwood Mountain Resort in Ferguson, NC which is a 1+ hour drive from Moses Cone.  They have their own riding trails, although I have never been so I can't comment on amentities or trail conditions.