I really don't want to beat the proverbial dead horse, so I'll just toss in a web page on the topic from the TSA. They now say, "Don't use a TSA lock to secure your firearm." (Well, OK. I'll give myself a break on my answer since this info was issued two years after I retired. )

They also have a list of 10 TSA approved gun cases. Two of those are "Snapsafe" cases made to go inside another suitcase. However, if you scroll down to "What kind of gun case and lock do I need..." the photo example is one of those blue plastic hinged boxes that some companies put their new guns in. Approved? Well, sorta. It's hard-sided, and you can put locks through those little plastic loops. So yep, you can use one, but they aren't all that secure. As I've said, inconsistencies....

Lastly, there just isn't anything in there that says "hard-sided suitcase." The TSA doesn't require it. If you think they should, tell them. Otherwise, we're arguing semantics and it's not getting us anywhere.

This topic comes up periodically on every gun forum. It's appeared a couple other times that I'm aware of on this forum over the last 4 years or so. It's to be expected, though, because it's confusing, whether it's your first time or you fly all over and the rules are just different enough in different places to be annoying. And no one wants to run afoul of a reg that'll cost you an $11,000 civil forfeiture for a single violation (and that's separate from any court matters.) Heck, even TSA says, "Different airlines and airports may have slightly different rules regarding the safe packing of a firearm and its ammunition, as well as the legality of traveling there with a gun in the first place, so be sure to read the airline’s policy carefully." So even the TSA acknowledges inconsistent application of some rules.

So there you have it. "YMMV " has never been more applicable.
Air Travel with Guns | Packing, Locks, Cases, and TSA Security Checks