e.g. could PRC or Hong Kong post office (which might be lower overhead operations compared to USPS) open a location in San Francisco in order to provide lower cost mailing/shipping to China as well as possibly other foreign locations on the Pacific Rim? Or are there traditional laws/regulations against foreign-state-run post offices?
I don’t see how treaties would be relevant unless the U.S. had a treaty with the foreign government allowing such a post office to be set up. What would govern the situation is United States domestic law, and under that the USPS has a monopoly on providing postal services within the US. That would stop such a thing being done, either by a business organisation or by a foreign government.
In the US, federal statute gives the USPS a monopoly on postal services. So no foreign state could open a branch post office in the US, unless federal law were changed, or unless there was a treaty allowing them to do so, but no such treaty exists.
There are a multitude of multilateral treaties that govern international postal conventions for sending international mail. See the Universal Postal Union and Cecil’s fascinating column on the subject.