Something which is quite likely being overlooked is that the priest, apart from having an obligation, may have had everyone’s best interest at heart ; even though it doesn’t appear that way.Furthermore, at a later date, if Amy decided to seek child support, there would be a record to refer to .
Or perhaps no one is considering what might happen if the child would later wish to discover who his/her biological father was , even though emotionally, these kind of things can sometimes turn out to be a “storm in a BIG teacup” . . . or worse. I think the child would still be entitled to this information about his/her own parents if he/she sought it.
Assuming all the facts are accurate, consider the priest’s position a moment: The only proof he has to go on that the father has absolutely no wish to accept any responsibility at all for the child , is the mother’s wish that the father have absolutely nothing to do with her and with their child-
The easiest way to verify what she is saying, would be to speak to the father – without coercing him in any way. It is possible that the bishop might wish to do the same thing only to substantiate or to answer what are reasonable questions.
If Amy doesn’t want the father around at all, that’s one thing (and more than understandable if he’d tried to get her to abort their child in the beginning - that’s an emotional hell for the mother). But how can a mother arbitrarily decide that the father totally and exclusively “is not part of the baby’s life”, at (the very) least biologically ? A fact is a fact. It appears, judging by the information available that the parish is only interested in the biological fact of fatherhood for their records ; but that Amy feels disclosing this information might/would somehow serve to draw the father into the situation.
People change and situations change, especially
In the realm of situational ethics, canon law is not exclusively cut and dry . But it provides the best guidelines and principles to help us orient our moral decisions and to understand our moral obligations. It is somewhat difficult to say what canon law recommends in this case due to the lack of availability of information . But it isn’t always required in every case, depending on related factors, to inscribe the name of the child’s father in Baptismal Records. Here is the most pertinent canon (877 §2 highlights mine)
Can. 877 §1. The pastor of the place where the baptism is celebrated must carefully and without any delay record in the baptismal register the names of the baptized, with mention made of the minister, parents, sponsors, witnesses, if any, the place and date of the conferral of the baptism, and the date and place of birth.
§2. If it concerns a child born to an unmarried mother, the name of the mother must be inserted, if her maternity is established publicly or if she seeks it willingly in writing or before two witnesses. Moreover, the name of the father must be inscribed if a public document or his own declaration before the pastor and two witnesses proves his paternity; in other cases, the name of the baptized is inscribed with no mention of the name of the father or the parents
More on Baptism and Canon Law at this link vatican.va/archive/ENG1104/__P2Z.HTM
Maybe someone with expertise in the legal medical field could weigh in here. Isn’t a person required at a hospital where they give birth - to also provide the name of the father of the child for their medical records ? If so, I believe that medical record constitutes a public document. If such a document exists bearing the father’s name, then according to Can. 877 §2 , the name of the father must be inscribed in the Baptismal record as well. The pastor of the place where the child was baptized would seem bound to try and fulfill that obligation.
Imagine for a moment, if you will, what kind of a mess there would be if everyone kept incomplete and/or sloppy records of births and incomplete records of Baptism.
Hope this helps. . . praying for Amy and her child.