All of this advice is much appreciated. I'm learning first hand what we are up against as gun owners. I wrote an appeal to the original denial letter, which is part of the process. It references passages from USPS's very own firearms shipping regulations plus some extras from the ATF website. Hopefully the claims reviewers will understand and/or believe their own regulations when presented in simple terms. Seems pretty clear to me but I'm not holding my breath. If this doesn't work I'll likely contact an attorney. I'll provide updates.
The gun was sent registered mail.