What if the priest tells one of them to bring the child back, and, as the father, have it baptized.
That sounds pretty good. But...?
I honestly don't know what is the "right" answer.
What if, in the meantime (while the "two" "daddies" are negotiating frowningly about which one of them is going to be "THE" official father), the priest goes to their Facebook page and sees they are planning a "Baptism Party" with gay-themed cake and decorations and party supplies, and displaying, incidentally, the lovely baby baptismal gown with rainbow sash and frills.
There's two Christian Gay Daddies who posted their children's baptism pictures at a gay dating site (Link)
Oh, it's a theme you can have a lot of fun with.
You might wonder why I'm going after this "gay" detour, but it's relevant for this reason: people are increasingly splitting off the Sacrament of Baptism from the Church, and using it as yet another way to celebrate "themselves".
Just as they've split off Matrimony: all self-expression and self-satisfaction, no sense of being part of a sacred Community which has definitions, doctrines, and requirements.
Straight people do this as well as gay people, of course. All personal interpretation and personal choice --- the overall values are borrowed from consumerism, not the Catechism --- no doctrine, no disciple; everything allowed, everything "on demand" like an entitlement or a public utility, nothing required.
Do not take this as directed to you and your loved ones! It's just what we see all too often in the parish: self-centered congregants are astonished when the pastor won't say,"Have it Your Way," "The Customer is Always Right."