The child is a minor and although they are divorced, he has two parents. While the mother has expressed no concern with his conversion, it would be the wise person who obtained the father’s consent (notice I do not use the word “permission”) before proceeding.
There is an old phrase: you are born into your family, but you pick your friends". That does not exactly translate to this situation, but I have not seen very many divorces (and I have been a divorce attorney in the past) where everyone is amicable. It is not necessarily that the father has any particular say legally (although without actually seeing the decree, I can simply say that is a possibility).
But beyond issues about the decree, there are issues between the father and the mother; issues between the child and each of his parents, and potential issues between one or the other parent and in laws/ex in laws. There have been numerous studies of the effects of divorce upon children (teens and pre-teens at the time of the divorce) and I believe it was Judith Wallerstein who tracked children of divorce and found that even in their 30s the children had unresolved issues; one in particular was the longing, still, to have the parents reconcile.
The short of it is that without addressing the father’s potential position in the matter, you risk putting the child into what could be a difficult, if not untendable situation in his choice to join the Church. If the father has no objection, then there is no issue. if the Father has an objection, then having the child wait until he is of majority is a far wiser decision which does not put the child in conflict with his father.
And the short of it is that I don’t care who was at fault in the divorce (and I am an absolute believer that one person cannot have a fight - it takes 2 - meaning that how they got there is not a particularly worthy exercise in trying to establish fault). The bottom line is that children often have issues which they may not make known particularly to anyone; there has already been conflict in their lives, and further conflict should be avoided.
I am all for him entering the Church. It may be far wiser to wait a few years rather than have this occur now, if the father is against it. As to who should contact the father concerning this, it should be left to the mother to ask the question unless you have a good continuing relationship with him. And if the father is against it, the matter should be dropped for now; there have been enough disagreements.