What you propose to do is legal,,and UPS allows for a non-FFL to ship a firearm if it's ok by Fed, State & Local regs.

UPS regs say you don't need an FFL to ship in some instances as allowed by law. The last sentence of their 'shipping firearms' reg is:
".....and from and between persons not otherwise prohibited from shipping firearms by federal, state or local law and when such shipment complies with all applicable federal, state and local laws applicable to the shipper, recipient and package."

UPS: Shipping Firearms


Getting them to even admit that it exists in their book of rules, much less someone there that understands what it is says is another matter. Plus you'll most likely have to show them that by current law it's legal to do the shipment .

They have 'FFL only' on the mind when you walk in there generally.