Most of the "lyrics" of the Ave Maria are from the Bible & should not bother any Christian: "Hail Mary, full of grace; the Lord is with you." [Ave Maria, gratia plena; Dominus tecum] -- spoken by the angel Gabriel to Mary at the Annunciation.
The second line: "Blessed are you among women; and blessed is the fruit of your womb" [Benedicta tu in mulieribus; et benedictus fructus ventris tui] is spoken by Mary's kinswoman Elizabeth, the mother of John the Baptist, when she greets Mary, who is coming to stay with her during her pregnancy.
In the book of Luke, where these stories are told, Mary goes on to proclaim the "magnificat" -- My soul magnifies the Lord/ Magnificat anima mea...
The very end of the Ave Maria might bother very strict anti-Marian Protestants, since it is not Biblical and was built up from various prayer traditions, ending with "pray for us sinners now and at the hour of our death." Still, it is ignorant to assume that the entire text is Roman Catholic.
Re: the organist's refusal: He wasn't being asked to pray the end of the text out loud personally, and his attitude toward playing the organ for the deceased and the bereaved shows that he's in the wrong business! I'm writing this on the anniversary of my own mother's death, and writing it as a church musician myself, as well.