Rules for flying with firearms are federal (under TSA) and not specific to nashville. you can google TSA's instructions. As best i remember, it must be in a locked case in your checked bag or a hard case of its own (more for a rifle) and ammo cannot be stored in a mag. Ammo can be in the same case but it has to be in the boxes it came in. I put a gun lock through the pistol as well last time. I also printed the TSA firearms guidelines and brought them with me which was a good idea because the employees were asking each other questions and didnt know.
Ill add that you dont need a $200+ case to fly with a firearm, just simple, $25 hard sided pistol case is fine, with a free gun lock from your local PD. (just walk in and ask for a gun lock). If TSA decides they want to look at your firearm, they are going to no matter what. They'll either cut the lock, call you to open it or pry the darn thing open, who knows. I like flying with a firearm in my checked back because it gets extra special treatment and always comes out first.