The original question is a mess - it's a mix of how to mail it to Alaska, mixed in with random TSA stuff.
Let's separate these two options out.
Ship to Alaska Option:
First off, a non-FFL holder can not use USPS to mail a handgun. You have to use UPS or FedEx, and usually you have to both go to an actual company shipping office (not the strip mall Mailboxes USA style affiliates), and then use some high priced 2nd or Next day shipping. But since all this is company policy not law, what's enforced and what isn't varies from time to time and store to store.
As stated, I don't think a non-Alaska resident can fly up there and receive a shipped handgun from an FFL. It's my understanding that you need to be a resident of a state to buy/transfer handguns to you. Now he could ship it to himself - but then he's "possessing the gun in NY to ship it, and local outfits there might want to see his NY pistol license before they ship it for him. Plus once it's up there, he can't mail it back to you, as I'd think a NY FFL would want to know why you're receiving a pistol that's already on your license.
Fly with it option:
This is where the TSA stuff comes in, not with the shipping of it. As above, if your son doesn't have a pistol license, he can't fly with it under stupid NY laws, and they would let an armed robber out of prison gladly to put your son there instead. So he can't travel out of state with it, and also would be dicey brining it back into the state.
There is one solution, which both solves the problem and does well for you. You tell your son that he needs to pay for you to fly with him to Alaska, so that you can take the handgun with you, and then leave it there with him. Then fly up there when he leaves and bring it home. That way you get a free Alaska trip out of it too.
Now just tell him he can't lose it, as I'd suppose on the stupid NY license you have to account for all your guns, though you could say you'd lost it yourself while up there.