Fallacy of composition. It would be factual to say that the USPS is part of the federal government, but then the statement becomes completely irrelevant.

And no, the USPS is not in the Constitution. Article I Section 8 gives Congress the power "To establish Post Offices and post Roads". They executed that power via legislation. The USPS didn't come into being until 1970, replacing the United States Post Office Department, a branch of the cabinet. The legislation creating the USPS says that it "shall not make any undue or unreasonable discrimination among users of the mails" -- what Trump has tasked it to do is in direct violation of that law.