Now THAT'S a great tour route!
Once again, Planner Extraordinaire AZBuck has posted a masterpiece!
Based on recent personal experience, here are a couple of additional ideas and/or areas of emphasis:
Not too far west of the Blue Ridge Parkway (BRP) is Damascus, Virginia. There you'll find the intersection of the Virginia Creeper Trail and the Appalachian Trail (VCT and AT). The VCT offers a 17 mile all downhill segment which the great majority traverse by rented bicycle. If your older child is a cycler, you can find a rental of appropriate size for him/her. I have many times seen smaller children in "car-seat" type bicycle seats or in trailers towed behind parent's bikes. This is not a hard-core mountain bike ride by any stretch of the imagination, instead it's a very easy downhill "glide" on the Virginia Creeper railroad line (out of use since 1977 and now solely a trail). I'd suggest doing the ride on a weekday, however, as weekends can get a bit crowded. Booking your shuttle up the mountain and the bikes in advance is recommended, too. Several bicycle liveries/shuttle services are on the 'Net. Getting to Damascus from Greenville, SC is an easier drive if you just stay on I-26 to Johnson City, TN, thence US 321 to Elizabethton (locally pronounced eliza-BETH-tun), thence TN 91 and 133 through the beautiful Shady Valley to Damascus. The shuttle and the downhill ride will occupy a solid 1/2 day, and at that time of year, I'd do the morning side of it, avoiding afternoon heat and thunderstorms. You can return to the BRP by either going through Mountain City, TN to Boone, NC, or by going north up to I-81 then a short jog to the south or east at many points starting around Wytheville, VA.
On the other end of Virginia, AZBuck mentions Assateague and Chincoteague and the Wallops Island aerospace museum. Assateague and Chincoteague are essentially the same place, with Assateague being a Federal National Seashore having limited (or nothing?) in the way of overnight accomodations, and Chincoteague having much in the way of motels and campgrounds. Chincoteague is a small island within the calm waters of the sound and Assateague has a large and well-developed ocean beach, complete with bathrooms and lifeguards. At Assateague you can drive your car right up to the back side of the low dune separating the parking area from the beach, minimizing the distance one's chairs, etc, must be carried. Bicycles RULE at Chincoteague/Assateague, with wide, paved bicycle paths providing safe travel from the village, across the small bridge, and onto Assateague Island, and from there throughout the National Seashore. There is a very nice (and, SSSHHHH!- educational) Park headquarters/museum on the island. The village of Chincoteague is readily accessible on foot, bicycles, or rental two-person "scooters" (and a pair of these would likely be seen as a real thrill for the kids to ride in). I believe the scooters can travel the 2 miles over to Assateague, too. Although I didn't have time to stop, the aerospace museum at Wallops Island looks like a most worthwhile stop.
A word of caution about crowds: The Chincoteague Volunteer Fire Department still holds the annual Pony Roundup in the summer. The third weekend in July rings a bell, but it's easy enough to confirm that. There is no getting to within miles of the place that weekend, as tens of thousands of "Misty of Chincoteague" fans flock to the roundup and auction. Nice, of course, if that's your goal, but a nightmare if not. In addition, the proximity of Chincoteague/Assateague to urbanized areas of DC/Baltimore/Philly translates to significant crowds on other summer weekends, too. I was there at mid-week last August and was rather surprised at how "well attended" the town and beach was.
Have fun planning and taking your trip!
Foy