Can it be done? Yes. Would I do it in your rig? NO!.
Once you leave Waynesville, 276 makes a very hard climb with lots of curves up to the Blue Ridge Parkway, then down a very steep climb to Hwy 64 in Brevard. In a car, you could easily run over to Brevard for the day and have an enjoyable trip.
If your are going to move over to Brevard with the rig, strongly recommend using I40 to I26 to Hwy 280. Very nice dual lane road with only a few smallish climbs.
My experience is in a Jeep (fun), single axle dump (wasn't fun), 35' FW (but had driven it plenty of times before I attempted, was during the week, and had the trailer in the other lane often), and truck camper.
If you do drive over and are at the top near lunch or diner, go north a couple of miles on the parkway to have a meal at Pisgah Inn which has fantastic views.
2015 Ram 3500 4x4 Crew Cab SRW 6.4 Hemi LB 3.73 (12.4 hand calc avg mpg after 92,000 miles with camper)
2004 Lance 815 (prev: 2004 FW 35'; 1994 TT 30'; Tents)