Make sure your lock is TSA approved. I've traveled with firearms many times and had no problems. I did learn and follow the procedures exactly.
[edit] Always put a bright colored chamber flag in each firearm. The TSA inspectors LOVE to see those. The flags make your inspection time short and quick.
no, in my experiences the locks on the pistol case itself cannot be tsa approved, i.e. only you can have the key or combo to the locks, tsa locks can be unlocked by someone with the appropriate key. in any event i always travel with extra sufficient tsa and non-tsa locks on me to account for any vagaries. know laws at departure/transit/arrival airports,
https://www.handgunlaw.us/
i very often fly with declared & checked pistols, including to/from/through even gun-unfriendly ord, phl, san, pdx, sea, except never/never to/from/through ewr, lga, jfk, bos. ensure that the pistol case is hardsided and has two locking points (pelican cases are gtg). apply non-tsa locks to the pistol case, and then place this locked pistol case inside some regular airline bag with a tsa lock applied to the latter. pistols and mags must be unloaded, ammo in factory box or plastic purpose-built ammo case only, all inside pistol case. do check in early and declare to airline staff, then follow airline & tsa instructions (each can be slightly different) and you should be retrieving this specially redtagged checked bag not on carousel but at airline’s own baggage service desk at your final destination. keep it all locked and closed up until outside the terminal at least. be patient, ime some airline and/or tsa staff are unsure and even perturbed with firearms (sea), others are a sheer pleasure and make small talk about their own pieces (las).
of course none of this explanation addresses the mechanics of lawfully transfering a firearm from the estate of a decedent, or by living person, including a relative, to another person who is a resident or non resident of the state in question.