I think it's not that the actual birthing is over, but that the pair is over. If the sentence read "The pair have one child together," it would sound like they were still a pair.
I think singular is more technically correct, but it just sounds too weird -- "the pair has one child together."
Had is past tense, which probably implies that the pair is over, they had a baby, and is correct since pair can be singular or plural (in this case singular).
Has, by itself (not as a helper verb) is present tense, and would imply that they are still together. The same with have (which is plural)..."The pair have a child together" would imply they are still together.
So, probably, all in all, "The pair had a child together" is the best usage.
Funny, they can get their grammar correct but not their facts! LOL
L
(PS, thanks to Louise who assisted me on this grammar consultation)
PPS, I am sure the non-native English speakers who regularly visit this thread (Chrissi, Snavel, Belbmmfan) will really appreciate this impromptu grammar lesson! LOL