There are so many places in the area, it is hard to say where to start. But, I would begin up north near Boone and Mountain City in the area of 421. In that area you can do the "Snake" and the "Back of the Dragon". Then I would travel south on the BRP down to Little Switzerland and spend one night there (you will pass some great places on the BRP like the Aquaducs on the way there). The next day I would do 226a (The Diamondback Loop) and then continue down 221 south to Hwy 80. Take hwy 80 (the Devil's Whip) back up to the BRP and then continue south. Go south until you get to the ridge along hwy 19 and either go into Cherokee or Maggie Valley to stay the night. (another option would be to exit in Asheville and do the Rattler before heading into Maggie Valley to spend the night.) The next day I would get up early and do a trip over to do the Dragon and the Cherohola Skyway. Also don't forget to do Moonshiner 28.
There are so many places to go, that you can only hit a few, but those would be my favorite places with your short time frame. Everyone talks about how bad the traffic is on the Dragon and how you should avoid it. I say, go do it (unless it is on Saturday or Sunday)! If you do not, you will regret it and then hate yourself. If traffic is heavy, just wait a while, and it will normally slow, that is when you go do your run.
Most important of all, HAVE FUN!