I've always gone to a Fedex center to ship guns as the Mail Box Etc. type places won't take them. I don't recall that you have to disclose what the contents are on the shipping form. When I've verbally told them it is a gun and I noted that they they don't mark it as such (they shouldn't - it would say "steal me"). Maybe they type something in the computer to note that but I doubt it.
Does Fedex have a certain amount of free insurance per package? And if you want more you have to fill out a form and disclose contents? I was thinking that if you disclosed it was a pistol and it was automatically covered up to say $500 & the clerk let you send it 2nd day, then you could just say was some piece of equipment valued at $500 if the package was lost. Even if they found it eventually, I don't think they would open it and you would just have to return what they reimbursed you.
Edit - just thought though that the only problem with this theory is the name of the place you're sending the pistol - Smith & Wesson, Glock, etc.
Some name aren't as obvious - Kel-Tec CNC wouldn't leap out to most folks as a gun manufacturer.