Before I get into this, I need to ask when Daniel was born. I do have a Daniel born Sept 1895, parents Turner Monroe Bowdoin & Martha, born Feb 1876, I do not her maiden name.
I need to correct some of your facts. You had John as Gov of NY, actually it was James who was Gov of Mass. This James was the son grandson of Pierre Baudouin our first in this country, arrived ~1686. Pierre was a French Huguenot who fled France due to religious persecution. He left with his wife and four children. I suggest you start with a google search for James Bowdoin, you'll get plenty of hits. Also, search a book site such as Amazon, there's a good book on him.
You're correct that the school is named after the Gov Bowdoin. It was James' son, also James that donated property, funds, to start the school.
There are family reunions every year that take place in Coffee County, Alabama. If of interest, sleet me know.
FYI, I am 13th generation in Pierre's line.
All for now.