You won't have any problems with any of the tunnels despite some being listed at 10-6". The clearance is measured at the white line and still going more up then in. So when you hug the yellow line, you will have another 2' in height at the edge of your camper and more by the time you get to the center where the AC is located.
I think the prettiest section is from Sylva to Asheville and then the section from north of Asheville through Craggy Gardens up to a little past Blowing Rock. Of course, all sections have their beauty and all are worth the drive.
Most of the overlooks you can stop at with the trailer in tow can you can sight see as you go along. You won't be able to stop at all of them due to crowds in a few spots, but plenty of other things to see.
With your size trailer, you will be limited in camp site selection so you may want to try and make reservations. Mid week shouldn't be a problem, but Friday/Saturday will be. No power but the nights usually still cool off nice.
I've been traveling the southern portion (Linville south) for 50 years and have done the entire route including Skyline Drive a dozen times in the last 20 years. Smallest rig was with my current TC, but I've also towed a 35' FW and 30' TT from end to end. Have stayed at every campground except Peaks of Otter.
Easiest place to pick it up going north would be east side of Asheville off of I40. Next easiest and the place I would really recommend is take 19/23 off of I40 near Waynesville up to the parkway and then north. This works well if you can get a reservation for Pisgah Campground, Loop A. Otherwise you may want to spend a night at the I40/276 just after you come out of the Pigeon River Gorge (Pride Resort or Creekwood Farm) so you can get an early start on the parkway.
If you want to experience all of it and are very comfortable with steep mountain roads, you could go past Gatlinburg, over 441 (15 miles up, 15 miles down). This route is not recommended for most with MH or trailers in tow, but if you know how to down gear and not loaded near max, it's a nice drive.
You should also check out BlueRidgeParkway.ORG which is the best site about the parkway. It includes a trail guide which descripes the trails at each overlook listed by mile marker.
Enjoy your trip regardless of how ever you decide to do it.