I checked the Chicago Tribune database, and the most recent mention of the Pullman Athletic Club is 1923. Their big event was an annual rowing regatta, and they also were involved in track and field, boxing, and football in the 1890s-1900s. Pullman was a company town that rigidly controlled all aspects of their employees lives. A gambling club here is VERY unlikely.

The Pershing Athletic Club, however, was a known gambling joint and a hangout for gangsters. It was around ca1926-1929 at least. It's armored door, guards, a peephole, and protection from the police, are all mentioned in period news articles. So this is a much better bet. The place was at 219 E. 43rd St., Chicago at the Indiana Theatre Building. The owners were Thomas Malloy and Edward Cohen.