Neither FedEx or UPS will allow you to ship firearms without an FFL, and a special account and signed agreement and since it is technically a handgun, then USPS is out of the question too.
I think you have two options:
1. Have the receiving FFL (if they have the special shipping account) create a label for you, then you can ship UPS no problem.
2. You can use a shipping service like ship-my-gun. This is how I have been shipping firearms that I have sold recently. It is a service that is offered by Buds gun shop that allows you to use their UPS account to ship firearms, and the plus side is that it is typically cheaper than it would be had you gone to UPS yourself.
Being a gun owner gets harder every day.