You know I had to get a CC. Love the look of this coin. This looks a lot like the 1880 CC I recently picked up.
Nice looking coin. I would roll a Q-tip dipped in MeCl2 or Acetone to get the gunk off. You could even be a little rough with it without hurting the surfaces. But it will get rid of the crap on the rim on the reverse.
Another nice CircCam. I agree that you could (cautiously and conservatively) work on the rim stains with a Q-tip, I think.
Before I did anything with a qtip I would just get a small jar of acetone and sit it down in it. Some stuff will just lift right off. I do this on all circulated raw coins i get. That will remove any surface contaminants. Even stuff like skin oil you don't see.