Don't buy TSA locks. Seriously, don't.
Every time I check my rifle, its a bit different process: sometimes the gate agent wants to see it, sometimes they don't care. They sometimes ask about ammo (11 lbs rule thingy, though they've never weighed the ammo by itself). You typically need to put their "This is a gun" signed form inside the case (at least on Delta, that's 100%, not sure about other airlines, they have their own polices).
After that its the mostly the same. Take the luggage to over-sized baggage, have TSA inspect it which requires you to unlock your combination lock, then lock it yourself and spin the dials. At no point in time should the TSA ever have access to the gun without you there. I believe one time I didn't even need to open it up for the TSA, they just x-rayed it. If they need access and you're not there, they'll probably call your phone number on the case. If you don't answer, they'll cut your locks. I've never had this happen because I've always watched my rifle case go through the system and confirmed with them that its a-okay.
Read the bold point again.