My only problem with all that is that Congress has already declared what the law is regarding just plain folks carrying firearms into "federal facilities". The USPS regulation purports to operate "notwithstanding any law to the contrary", placing themselves above the Government of the United States as supreme sovereign, able to dictate what's what irrespective of what Congress and the President have enacted as law. The USPS doesn't really have the authority to dictate terms to non-employees that exceed the scope of the law. When a statute expressly states the terms that apply, the presumption is that Congress meant what it said and that the President endorsed it. If they meant to allow greater intrusions into our freedoms, they'd have said so. "Expressio unius alterius exclusio est." (The expression of one thing is the exclusion of all others.) Except where Congress has specifically delegated legislative authority to an agency by law (e.g., the Pure Food and Drug Act), that agency has no greater power than the President to tell folks what to do or how to do it. That's why the USPS has no authority over guns in the public parking lots.
The buildings, however, in which U.S. employees routinely do their work, are "federal facilities" under 18 USC 922, so, when there's a sign conspicuously posted or for some other reason one has actual notice of the prohibition, he can be convicted of a crime for carrying a firearm into the post office itself.