If you do go to Lake Cumberland SP, try picking up I-64 off I-75 to Rt 127 south instead of using I-75 all the way south. Takes just as long but is easier driving at 55 and 60 with 4 lanes at least 2/3 of the way down Rt 127. You will pass within 5 miles of Four Roses Distillery with tours Mon-Sat 9 to 3 on the hour. Very informative even for kids.
From LCSP, take Rt 55 north to Lebanon, KY and tour Independent Stave Barrel Co. Call them first as sometimes the tour is down. You will pass Green River SP on the way and you can check out that CG and see if it fits your style. LCSP does have shaded spots, maybe not full shade but certainly morning or afternoon shade as the campsite area is surrounded by mature trees.
I often camp at My Old KY Home SP in Bardstown, KY. It is not on a lake, but is surrounded by many historical spots and distillery tours. Great eating in nearby restaurants including Mammy's Kitchen in downtown Bardstown and the Whistle Stop restaurant in Glendale, voted the best non-franchise restaurant in KY several years running.
Have a great adventure...