As to the poor sap named in the post, he’s making his bed, and deserves to sleep in it. If he signs on the dotted line, he’s screwed. It sounds like he knows it. He should put his foot down, but he won’t. This marriage will end in divorce, and this purchase will end in foreclosure.
I’d say call her bluff now, and when she files for divorce, sue for full custody, no visitation, and child support. Because the alternative is finacial suicide, to make a person happy who is willing to use their own children to get their way. Such people can’t be reasoned with, whether they are male or female.
Full disclosure: I’m a guy. But this isn’t sex-based. My advice, such as it is, would be the same if the genders were reversed, and there are just as many spoiled, self-centered men out there as there are women. I’m fortunate in that my wife and I have had this conversation, and until we can buy something that we want, on a traditional loan, at <30% DTI, we'll rent. That model is coming back. I'm sure of it, because every time the RE industry has gotten away from it, it's been a disaster, and they've come back to it, because it's the only sustainable way to finance residential real estate.