I just looked up B. Edwards at Baseball Reference and found two minor league players by that name from the late 1920s. One was a third baseman in the Mississippi Valley League, while the other was shortstop in the Western Association. There's no certainly that your bat belongs to either, but it's a start. They might also be the same player. Each played a full season.
Old baseball guides like the ones made by Reach and Spalding might help. There is a ton of info about minor league teams in them, along with team photos.
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