Go to Cades Cove on Wed or Sat mornings. The road is closed to cars until 10am. I try to get there before 8am. There are some short but steep hills on the loop so be prepared for those. Watch out for people dismounting in the middle of the road and not expecting anyone else to be able to ride up. The road has just been repaved so it is quite nice. It's not a loop to speed around but to take your time, look at the views, spot the animals and enjoy being there.
You can try the Foothills Parkway but I would go some distance away from Gatlinburg. It has many climbs and no services so be prepared. I wouldn't recommend US441 through the Park or the Little River Road for riding this time of year. Just too many cars. I suppose if you got an alpine start it might not be too bad. Most tourists get stuck at the Dennys breakfast bar until later in the morning. The climb up to Clingman's Dome out of Newfound gap is a nice one, about 7 miles each way. Again, go very early and have a flashing light on your bike.
I don't know the Tenn. side roads all that well but my opinion of Gatlinburg, Pigeon Forge and Sevierville is not suitable for public consumption. If you drive over to the NC side you can ride up the BRP from its start just outside Cherokee. It's a long 10 miles climb from there with no water or services and 5 very dark tunnels to ride through. I do that climb on a regular basis though usually not in the summer.