I think there's a campground on Old Hickory Lake really close to Saunders Ferry Park in Hendersonville. I think I remember seeing it on 31E just past Saunders Ferry Rd, but I could be wrong. From there, Cedar Hill is 10 minutes, and Triple Creek is maybe 15 minutes away.
There's a small campsite at Seven Oaks Park. It's kind of tucked in close to holes 8,9,16 and the inner course, and there's only room for maybe 2 tents, but there's a picnic bench and a fire pit. I don't know how you go about reserving it, tho.
The last option would be Sharp Springs Park in Smyrna. It's a big park with several shelters. I'd be surprised if you weren't able to reserve a place to camp somewhere in there. Again, not sure how to go about it.