I've done some more digging, and I can't find anything connecting either the father or grandfather to baseball.
I also found the following photos at
http://www.attheplate.com/wcbl/index.html, which is a website devoted to Western Canada Baseball:
According to baseball-reference.com, Archie Persons, in addition to playing for Montgomery of the Southern Association from 1908-1909, also played for the Bassano Boosters in Canada in 1912.
Looking at these two photos, along with the lack of any corroborating evidence to the contrary, makes me more inclined to believe Archie Persons the baseball player was NOT either Archie, Arch or Archulus Persons who was related to Truman Capote.
Steve