OK. I bought a hard-sided lockable case for ~$25 and purchased a triple, same-keyed, set of padlocks for the case.

Arrived at the airport early, declared it, signed the document, and then the agent asked me to unlock the case so that she could put the form inside. She also asked me to show her that the shotgun was unloaded. I was shocked. I said, "it's a pump action Winchester. It's going to be pretty visible and loud." She said no problem.

So I opened up the case, lifted it out, and pulled that pump back hard and fast. She actually jumped a bit.

TSA had no problem with me locking the case before they scanned it. They said that the agent checking it was enough.

Anyway, I hope it makes it.