All the sites in the Smokies have to be reserved.
I would avoid Gatlinburg traffic if at all possible. It will be REALLY bad that weekend. Its bad almost every weekend.
Cades Cove is popular and your kids might enjoy it. Its a short 2-3 mile hike to either site 9 or 10. From there, its a straight shot up to the AT ridge. Sounds like just what you are looking for. But traffic around the Cades Cove loop will be bad that weekend. The kids might like to tube the Little River there at Townsend too.
You might consider the Cosby area. It has good access to the high ridge of the AT. The Albright Grove has some amazing trees. You are fairly close to Mt Cammerer too. There aren't any really close backcountry sites but the frontcountry sites there are better than most.
The Big Creek area is probably my favorite. You can go up the Benton McKaye trail to Mt Sterling and great views. You can go up to the AT and Mt Cammerer. Big Creek will provide hours of fun for the kids. There is no backcountry site closer than 5 miles though. You could car camp at Big Creek.
Map is here: http://www.nps.gov/grsm/planyourvisit/upload/trail-map_JULY13-full.pdf
One of the best things about the Smokies for my kids was the opportunity to explore along the creeks. And by Memorial Day, you'll probably appreciate a cool creek or a high ridge. Almost all the sites are on a stream.
Alternatively, you can consider Slickrock Wilderness. There is a beautiful stream, high balds, some good climbs, and no camping regulations.
