Not sure if this would help. In the census, Patrick and his wife, Sarah, are listed as being from Canada Eng.
They have in 1900
Mary E, age 15
Francis, age 14 (son)
Margaret age 11
Rose, age 11
Catherine, age 9.
Francis was born on 16 Mar 1887 and on his birth record there was a reference to that fact that Patrick and Sarah were from PEI. Not sure how much access you have to those, but it might help. Later reference in the census is to Nova Scotia...but Patrick's parents were born in Ireland and he was a naturalized American citizen...
If you contact Waltham NARA via e-mail (google for it) they might be able to provide additional information from Patrick's Declaration of Intent and Petition for Naturalization. It might lead you to exact town of birth and if he is indeed, the brother of Francis, might help you track him down.
You might attempt to find this Francis Anderson, son of Patrick, and his family. Maybe a child or grandchild of this Patrick could provide some info. Patrick was born about 1861 so this might provide a ball park age for the Francis you seek.
You might want to search those church records you found for the marriage and birth of Francis' child. If you check from her birth to 1891 for death records, you might find that the wife died in childbirth or something thereafter. Many times if you can check those death records by marriage or maiden names, you will find it in the same church where they were at other times. Just a suggestion.
If I uncover something else, will let you know. I did find a death record for a Francis Anderson, but parents listed were different then those you mentioned so must be a different individual.
Have a good day.