Obviously checked baggage, unloaded, and through the barrel gun lock.
And I dunno if I'd bother. They are so damn paranoid it may be better to see if you have family there who will loan you a carry piece.
Don't give info you don't know. no lock is required on the gun itself.
I've flown several times with a firearm and it needs to be unloaded and in a lockable hard sided case. Ammo can be in the case with the firearm but must be in a container of its own, like the factory cardboard box or a plastic box. It can't just be thrown in the case loose.
Declare you are flying with a firearm to the ticket counter person. They will have a TSA person come over and take the bag you have. Together yo uwill check the contents of the locked hard sided case. They will place a inspection tag in the box. you will then close the box, lock it and your luggage and the agent will take your bag and put it on the conveyer and moved out of customers site.
The bag is to never be left unattended behind the ticket counter. IT must be sent behind the "wall" (Where another TSA person can steel it)