I deliberately made no reference to gender in my post - makes not a bit of difference (though birth certificate comment means that it pretty much has to be the dad).
Think about it. This person has seen this baby ONCE A WEEK so far. Once a week. That isn't anywhere near enough for the baby to have any kind of bond with the person. To the baby, this isn't yet a parent. It's a person whose touch, smell, voice they HARDLY KNOW. It's a virtual stranger.
That's not what this very 'my rights, me me me' so-called parent wants to hear, but it's a fact, sadly.
To leave the baby overnight in the care of this parent would be distressing for the baby at every level, because not only would the baby not really know who they were (and would be mightily upset at the disappearance of the primary carer) the parent wouldn't - can't - really know the baby: their routine, their personality.
This parent needs to realise, and very quickly, that parenthood of such a small baby is nothing at all to do with RIGHTS and everything to do with RESPONSIBILITIES. Their behaviour so far (don't want to go on birth certificate, but do want to play dollies with baby) is downright shoddy. And doesn't inspire much confidence in the future, sadly.
I would consider it the RESPONSIBILITY of the parent with care (i.e. the one who knows the baby because they are the one there putting in the hours!) to refuse this needy request, IN THE INTERESTS OF THE BABY. It is in no way good parenting to hand a four-month old baby over to a person who is a stranger simply for the purposes of indulging them.
I imagine that the second parent won't like this at all, and shouts of 'It's just as much MY baby as yours' will ensue. But I'm afraid that that's simply not true. Let's go back to RESPONSIBILITIES not RIGHTS. Who here has the responsibility? Who is the one giving, not wanting to take, from this baby?
I would want to see a far higher level of committment to GIVING to this baby before I could take this parent seriously. The fact that they are even asking for overnights with a baby they see once a week is worrying enough and indicates that they have no perception of the baby's real needs - just their own. Until that changed, like I said, they could go take a hike!