Jefferson Memorial has plots you can reserve and trails. It’s not LBL but easy drive. Aside from camping, great for day hikes. That’s in Fairdale.
Deam Lake in Borden, Indiana is about a 30 min from downtown and still in metro technically. Bridal paths surround lake but on the side of the lake opposite the public park there is a great camping spot on a little peninsula. The Knobstone Trail cuts through Deam Lake. That’s a long trail. Pretty sure it goes all the way to Indianapolis.
Otter Creek Park near Fort Knox is pretty great for camping. Park has a lot hidden. Not necessarily straight forward where to camp, but there are great spots.
Not sure if Big Oak in Madison, Indiana allows camping but that’s a good day trip and can try a few of their breweries on way out.
A bit further north not far from Bloomington, quite a few places to camp and hike.
Others:
Hoosier National Forest
O’Bannon Woods (near Corydon)
Bernheim Foest (Bullitt)
That’s all I have