"What novel requires a woman to take care of a child on her own without informing the man, especially when the woman is unmarried or divorced? It's utterly fantastical.
If the woman's family is wealthy, perhaps they could pull some strings, then deliver the child at an expensive private hospital. However, in real society, you first need to obtain a certificate from the community. With this certificate, you can then register and go to a public hospital for prenatal check-ups, schedule the birth, and register the child's household.
If you are unmarried or divorced, without a certificate issued by both spouses, you can't even register at a public hospital, let alone get a household registration for the child.
If the Lu Family is very wealthy, wealthy enough to disregard these tedious procedures, that's another story. Or maybe in some remote mountainous area where information isn't readily available, it wouldn't really matter. But in Yan City, it's nearly impossible.
