Montgomery Bell isn't that far from Nashville. DS lives a little past it and commutes to down town Nashville for work.
Suggest Natchez Trace State Park. Google puts it just under 2 hours from Brentwood but should be easy. Lots of gravel roads in the nearby forest for riding the bikes. Believe you can also use the horse trails in the forest.
Not a state park, but close by and free. If you don't mind dry camping. Meriwether Campground on Natchez Trace. It is at mile marker 385. Would be nice when the weather is cool.