The Waterbury Button Company is still in business today and have been since 1812, many Civil War military buttons bear their back mark, as well as today's military buttons, I think they are the only company that made CW military buttons still in business today.
Look on page 107 red book. "Large letters, post-CW" You are going to have a difficult time trying to find a pictured bm since you have a cuff button which are not identical design bm to the coat size.