I would never ship a modern or C&R handgun that way. Not worth the risk. I ship those fedex ground.
You're still not home free... while it's NOT illegal to ship a modern or C&R handgun by FedEx ground or UPS ground, it's clear that FedEx or UPS didn't know you were doing it. You're going to pay through the nose if YOU are shipping a handgun.
Federal regulations require you to declare to the carrier that you're shipping a firearm, and the counter person will almost always ask if its a handgun. Both FedEx and UPS counter people will then tell you that their tariffs require that a handgun be shipped by one of their premium AIR services.
Surprisingly, you can't take a handgun to be shipped to one of the UPS retail stores, either. You have to go to a UPS HUB, and those are often quite a distance away. FedEx requires you to go to their hub offices, too.
When shipping FedEx or UPS ground services, if you ever have a hand gun lost or damaged in transit, you'll probably NOT have a claim for your loss... even if you took out insurance.
IF you go to the UPS or FedEx web sites and do an inquiry about shipping handguns you'll find that they REQUIRE you to ship by Air. This is NOT required by law, but it is the CARRIER'S operating rules (called their TARIFFS).
(The requirement to use AIR services started some years back when UPS and FedEx were having a big problem with the employee theft of weapons. The AIR services have much tighter controls and double-checks. This let them cut their losses and allows them to charge their customers more.