I would trust FedEx for this -
But I would not trust the USPS, based on prior experience with other shipments, and the horribly rude and unhelpful people I have encountered at the USPS counter in Virginia Beach. (Surprisingly, the USPS people in Orlando seem much happier and much more helpful, perhaps it varies more by locality than I realized).
Even UPS "lost" a Glock that was being shipped. The box, which was shipped with 6 handguns in it, arrived damaged, and only contained 5. My FFL signed for it anyway. I wasn't too happy about that, but they helped me run the trace. Ultimately, the shipper, Hoffman's Gun Shop in CT, would have been out the $$, and we were working through that, when UPS "found" it in their warehouse in Richmond. I wonder how much truth was in the official story. Doesn't matter, I suppose, the Glock showed up a few weeks later.
I take a risk every time I buy a classic/collector piece from another store. It's a risk that has worked out more often than not. I've bought a lot of guns on GunBroker, and a few on other sites, all of which were shipped (via various means) to my local FFL. Many of them via USPS, and all of them arrived, eventually.
My bigger issue - dishonest sellers. Who lie, or hide information, on the firearm they are selling.
But that is a topic for another thread.