I know about a case of a non-Jewish husband who converted to Judaism while his Jewish wife was pregnant. The beit din rushed to finish the conversion process before the baby was born, and I don't quite understand why. I am under the impression that fatherhood (unlike motherhood) is determined at conception according to halakha, so that the father would not have a halakhic father-child relationship with the baby regardless whether he converted before or after its birth. I'm not 100% positive --- but if I'm right, why was it important for the conversion to precede the birth? Clarification would be appreciated.
You're correct. From a technical standpoint of Jewish law, at the time of conception the father was not Jewish and therefore no blood relation is recognized vis-a-vis the father.
There are practical reasons why it would be better for the father to be Jewish, for instance all the ingredients in the kitchen could be kosher, but in some cases the food isn't kosher if handled by non-Jews. Furthermore it's just better for parenting if both parents are on the same page religion-wise. Not knowing anything else about the case, that would be my guess what drove the beit din.
I'm told that if dad isn't Jewish as of this moment, some synagogues may recommend a quieter home circumcision rather than one at synagogue with the full pomp and circumstance (to avoid an implicit endorsement of intermarriage). That, or something as simple as giving them family membership at the synagogue, may have factored into the decision as well.