The oxidation is often accompanied by a sulfur smell. The smell can be removed without cleaning up the mouthpiece, but I'm going to do both. I know that I can polish for hours with something like Simichrome to remove the oxidized layer, but that is extremely slow. It is also possible to start with something like 1500 grit wet/dry sandpaper (using water) and quickly remove the oxidized layer, but that seems too aggressive.
I decided to try inexpensive fingernail polishing and buffing blocks used by manicurists. Type in something like "nail polishing block" on Ebay and you will see what I mean. They tend to have 4 sides with different grades of sand paper (it isn't even sand paper, it's way less aggressive). They are about $1 each and one can do several mouthpieces.
I first tried it just on the beak to see how aggressive the #1 side was.
Now I'm getting close.