I live about 50 minutes from the BRP. It's totally worth it. I'll be honest, I never considered the heights of the tunnels and never had an issue.
There's a campground right off of the parkway in NC not far past the Va. boarder going South called Racoon Holler. It's in Glendale Springs. It's a great campground but, make your reservations, they can fill up. If you stay there make sure to see the church's of the frescos. One is just about walking distance from the campground and the other one is nearby in Jefferson NC. Also, that campground will occasionally have live music on the weekends. Once when we were there Doc Watson's band was there (been a few years ago). They were awesome but, the best part was that they were camping a couple of sites down from us and we set around the campfire (as did many others) and enjoyed the jam session till late at night. Great stuff!
Also, get a meal at Mabry's Mill. Great pancakes and country ham.