There may be a connection, but I am not sure of it. William's son William married a woman by the name of Mary. She was 36 in the 1880 census. They also had a young daughter Mary who was less than a year old then. They lived at that time in Irondequoit.
The same census has a Michael and Anna Bauman in Rochester. He is shown as a tailor. I do not know if there was any connection, although it is possible he was a son of the first William. He is listed as from Germany.
My connection to William (if there is any) was that he may have been related to my ggggrandmother Margaretha Baumann. I am guessing based on his name appearing in both her will and her husband's will and based on age that he may have been Margaretha's brother.
So I'm not sure if we connect, but perhaps we can find some more commonalities.