IMHO, bringing a firearm in a secure package to legally* mail it means that it is not loaded or readily available to you (it's boxed, right?) and stays with the USPS when you leave. Logically, you have to "carry" it into the PO in order to legally mail it.
"Normal" carry, where you're wearing it on your hip (or elsewhere), and it's accessible to you while you're in the PO, and you carry it away with you again when you're done doing your business, is not necessary in the same way. I know, I've bought stamps etc., while unarmed.
IANAL, etc., etc., but IIRC one can normally do whatever is reasonably necessary in order to carry out an act that is specifically provided for by law (in this case, shipping a firearm). Otherwise the legal act is not really legal, and legal interpretation tries to avoid obvious contradictions.
That's a general legal principle, btw, and I do not say that there can't be some (unintended?) Catch-22 in some law somewhere, or that there isn't some fine point of legal interpretation that I don't properly understand. Comment from actual attorneys is invited.
Not trying to start up the "can you carry in a PO" debate
* I do not review the situations where it is legal to mail a firearm. Separate question.
** Yes, yes, I know all about the debate about carrying and "lawful purpose". I express no opinion on that question.