Congrats Bear Finder, I believe you've just joined a unique mystery club.
There have been other posting of similar or identical items I remember some, including myself, believed it to have been the top cover over a wine bottle cork. Other possibilities were Emerson Radio, Edison something or Eastman.
I recall that in that posting, someone had submitted a map where these things had been found, only about a dozen of them.
Dave, I like your concept of organ component. If not organ then perhaps some other kind of musical instrument.
Bear Finder, Look up posting by "D.C. Matt". I remember he had participated in that discussion and may have been the poster of the map. You might look for "E Button" in an advanced search.
I think the fact that the stamp is off center suggests that other buttons are out there stamped with other letters. It might be interesting to use the "Post Photo" feature in the Bing search bar to see if other examples might show up in other music notes/keys. This is the first time I've seen the context of a church yard associated and may be a significant detail. Perhaps coffin related.
Best wishes!
