The woman should survive. The baby should die because if he knows he is immortal, he will get overconfident and arrogant. The woman can also make more babies. In the POV of the husband.... A husband always loves her wife in a good relationship. The husband and child have not spent any time together which means less "love" for the child.
So the baby should die because it will become arrogant 'knowing its immortal', and yet the woman wouldn't? It feels like there's something of a gender bias. Not saying there is, just in the way you refer to the baby being a male and arrogant when the sex of the baby wasn't stated, and also how when you refer to the compassion of the husband you refer to him as "her".
On the larger case, I hold that it is ultimately the woman's choice on this matter, for most of the reasons stated before that it is her life to sacrifice. Not getting into the issue of sentience all the way through, sentience doesn't necessarily grant moral privilege.
And even granting that this child has the moral privilege, it still isn't a necessity under these terms. I will ask a similar question, which should demonstrate the issue I take with the position a mother must die to guarantee the birth of her child.
You see a man who is on the operating table and getting ready for a surgery, but the problem is that there is nothing to transplant for his heart. You can give your heart to save him, and only your type of heart will work. Or you can do nothing, and let him die. Must you save this other person? (Assume the victim isn't conscious and there is no way of communicating with him/her.)
Without taking the third option out, what is your answer, yes or no? And if you answer this question differently from the "Must the mother give her life for the birth of her child?", then what make one answer the choice of the person and the other a forced choice?