Yes, the Constitution requires that the federal government have a postal service and post roads, but it doesn't say that these have to be a MONOPOLY! How can a monopoly lose money? Only by being a government-run union operation! The same people who manage the (also going bankrupt) federal budget!
Simple solution -- eliminate the monopoly, let private companies compete delivering first class, certified, registered, and junk mail to your mailbox. Not as government contracts; a totally open market; multiple carriers in metropolitan areas; higher delivery rates in rural areas. If the USPS and their union can compete, so be it. If they can't, then they are Hostessized.
I know, I often dream a lot.