I guess I've answered my own question:
ATF says: "A nonlicensee may ship a firearm by carrier to a resident of his or her own state or to a licensee in any state. A common or contract carrier must be used to ship a handgun. In addition, Federal law requires that the carrier be notified that the shipment contains a firearm and prohibits common or contract carriers from requiring or causing any label to be placed on any package indicating that it contains a firearm."
But then FedEx says: "FedEx will only accept shipments of firearms when either the shipper or recipient is a licensed manufacturer, licensed importer, licensed dealer or licensed collector and is not prohibited from making such shipments by federal, state or local regulations when these conditions are met. FedEx will accept and deliver firearms between all areas served in the U.S. and Puerto Rico."
So, ATF says I can do it, but I must tell FedEx. FedEx Says they won't do it unless it's to a (or from) a licensed dealer. Now, suppose I were to try to slip one past FedEx, and then suppose something were to happen to the gun in transit. Do you think FedEx would pay on an insurance claim for an item that was shipped in violation of their policy? Or worse, did I just commit a federal crime by failing to tell FedEx I was shipping a gun? Can I tell them "Hey, I was complying with most of the law"...... as a defense?
UPS is even worse: “Firearms will be transported only between licensed importers, licensed manufacturers, licensed dealers and licensed collectors... to a licensed importer, licensed manufacturer or licensed dealer for the sole purpose of repair or customizing, and the repaired firearm or a replacement firearm of the same kind or type on return from the licensed importer, licensed manufacturer or licensed dealer to that person.”