I have NEVER had a problem shipping through UPS. Usually, I take it to a local hardware store which is a UPS pick up point. I'd have to drive 25 miles away to go to the UPS office/distribution center. It probably costs me more, but like I said, I don't have to drive 50 miles just to ship out the gun. I also know the UPS drivers that pick the packages up.
I have had one damage claim problem, but I didn't have to fool with it. I ordered a barrel from Rice, and he sent it out. Well, it never showed up. He said he'd take care of it, and just sent me a new barrel. He told me that UPS simply lost it. Had no idea where it was. A few weeks later, he told me that they had found the barrel and it had been bent over almost double! They paid the claim, though, no problem.
I suspect the easiest way would be to call for a pick up. When you call UPS, you are calling a central office, and if they have any questions, SURELY there will be someone there who will know what to do. I haven't done it in a while, but you will have to know the weight of the item plus the total size of the package in EVERY dimension. It's a bit cheaper to send to a commercial address than a home address, so if you can send to someone's workplace, it will help a bit.
I personally wouldn't DREAM of sending anything like this through the post office. I've been told it can take YEARS for them to come through on a damage/loss claim.