I've back country camped in the Big South Fork and it is a beautiful place - Angel Falls Overlook is my favorite. I don't know much about the campgrounds. Bandy Creek Campground has 2 group camp sites - looks like $100 for up to 25 people plus $3 per additional person. Sites have "cook shelters" and campfire rings but not a lot more info on the website.
A few campgrounds in the Smokies National Park have group sites but I have no experience with them either. I looked at Cades Cove and saw a site for up to 30 people w/ picnic shelter for $75/night. Pets are prohibited in some of the national park campgrounds but permitted here.
I looked at a couple state parks (Fall Creek Falls and Rock Island) between Knoxville and Nashville for comparison. Looks like standard state parks rates are $55-70/day to rent a picnic shelter and $20-24/night for tent only campsites.
Just throwing out those options... not trying to sell any of them.