GSM is a great choice! There are a few mountain-side tunnels you have to drive through in the park, very short, maybe 50 yards at most. Hope your daughter is okay with that.
Shame about the caves. I know it probably won’t help comfort her, but more people are going to die in car wrecks in Indiana today than have ever died from cave-ins at Mammoth cave.
Clingman’s dome at GSM is (or was) the highest point in the Appalachian mountains. Def visit in the evening if you want amazing views. Cades Cove usually has bears and is a driveable trail. We saw 2 bears and a bear cub, plus some deer and Elk on the drive.
I saw in another post that your fam like aquariums, there is the Ripley’s aquarium in Gatlingburg, which was a little expensive IMHO, but it is a decent aquarium.
Skybridge is a ton of fun, but very touristy.
There is a little GSM trail at the edge of Gatlinburg that my wife and I walked at night. We frequently do this in parks. Bring red lights to preserve your night vision, and a black light because there are tons of centipedes that glow a ghostly blue when hit with a blacklight. We also blinded the shit out of a few mice. Try not to do that…
Also, depending on the time of year, do some road cruising in the morning or early evening. You’ll see lots of wild turkeys, deer, and snakes. Please be careful not to run over the snakes.