To be fair, you can do things like protest on post office property. We have an itinerant group of Lyndon LaRouche followers in the area that will set up on post office lawns and have a small demonstration.
Only outside, only if they behave themselves, only if they're not asking for money, and only at the discretion of the local postmaster. He may be a Rouchie himself (though probably not).
They don't seem to interact with anyone unless someone asks them first.
That's probably because you are not allowed to solicit at a post office. By law they have to wait until someone approaches them. They may have learned that the hard way, but at least they learned it, unlike those "civil rights" auditor idiots.
I imagine that if it's a federal property, open to the public, anyone can protest there as long as they aren't causing issues.
It doesn't have to be federal property. It can also be a rented space. Some jackasses tried protesting in front of a post office inside a mall (this was back when malls were still a thing) and the postmaster banned them from the entire mall.
Post office laws are weird. For instance any bag or vehicle you bring onto post office property is subject to warrantless inspection
unless it's something you're there to mail. They can search your backpack, they can search your car, they can search your person, but they can't search the box of cocaine you're mailing to your Aunt Ethel (at least not until you've put a stamp on it and handed it over - also you do have to declare it).
Basically, if you're making an ass of yourself at a post office, there are all sorts of odd laws the postal employees can use to make your life miserable, even though they don't usually bother.
tl;dr: do NOT fuck with the post office.