Fascinating find, k-j! Mark Cooper has (or at least had) a very similar set -- the only other one in existence that we're aware of -- which is shown on page 30 of his Baseball Games catalogue.
His set, which he labelled "19th Century Bowling Pins," "Manufacturer unknown," likewise includes six wooden pins, with player images identical to those on the set you show, and a ball roughly three inches in diameter. However, the pins in his set are shaped differently -- thicker and more cylindrical, less tapered, and with significantly different top finials -- and the ball in his set appears more billiard-ball-like, possibly ceramic and definitely with no medallions.
The Brouthers-Clarkson features, then, are the first we've heard of in connection with this bowling set, although their appearance on The Champion Game of Base Ball (A S Schutz, 1889) is well known. If the ball in the set you show is authentic to the era, it might well date the bowling set to 1889, same as the Schutz spinner game, since that season (in which Brouthers won the NL batting title and Clarkson led the league in most every major pitching stat) was the only season the duo were teammates, and it would make sense that they'd have been used together promotionally.
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