Just know that some of the areas of the BRP and inside GSMNP the roads are NOT RV friendly.
Not so. Only a few of the pull outs are not RV friendly. I've traveled the entire length with a 35' FW with no problems.
I enjoy staying without hookups and will provide you how I would do it. Work you way to Townsend, TN and visit Cades Cove. If you can do without hookups, stay inside the park. Cades Cover is an 11 mile loop an the best place to look for wildlife. Then take 441 up and over the mountains and stay in Smokemont for a day or two exploring the east side. Either make a side trip or run over for the day to Deep Creek and hike to trail to a couple of the falls.
From Smokemont, it's easy to grab the southern tip of the Parkway. Your first campground opportunity on the parkway is Pisgah near Asheville unless you want to get off for a night. If you do, stay at Moonshine Creek. You could spend a couple of nights here and run down to Dillsboro to take a train ride if into that. On the parkway, make sure you stop at Devil's Courthouse and take the trail UP to the top, well worth the views. Park at the southern end of the parking lot if possible and you will be able to snap a birds eye view of your rig.
May want to drop off the parkway and enjoy Asheville for a few days before heading further north stopping at Lineville Falls campground. Spend a couple days here traveling north on the parkway to enjoy the Linville Viaduct along with hiking the Tanawha Trail as it's one of the prettiest. Parking is tight for the trails so you will won't to stop without a trailer in tow.
While you can continue up and stop at Doughton Park, that is about as far as I would go north on the BRP. Here you can grab I77 south to I-74 to I-40 through Raliegh NC and then catch 70 to Cedar Island or 64 to Nags Head depending on how much of the Outer Banks you want to explore.
If you weren't worried about the mid Atlantic Coast, stay on the parkway making sure to plan on a meal at Mabry Mill and stopping at Rocky Knob. You could continue up to the northern end of the BRP and then continue up the Shenandoah Parkway (It is a $25 fee for the pass unless you have an America the Beautiful pass). From the north end of Shenandoah Parkway, grap I-66 west to I-81 through PA and up around to at least Boston area.