You may consider looking into either Hillman Ferry or Piney campgrounds. These are National Forest Service cg's located at opposite ends of LBL with Hillman Ferry being the Northern one. I can't speak for Hillman Ferry but we have camped at Piney before our kids took the training wheels off the bikes and they are now in their 20's. Piney is a large campground but spacious with a lot of activities geared towards kids. They have summer activity programs, basketball, baseball, a nice sandy beach, two boat ramps, a fishing pier, and a paved bike trail off thru the woods. This is a very family oriented campground located on the shore of Kentucky Lake. The 1850's home place is a great day trip as is the planetarium. I have a few pictures from a Halloween trip we took 6 or 7 years back.
Piney It'll give you some idea of the campground.
"Onward thru the fog!"