Scenario:

A woman is planning to visit her American fiancé for a few months in the U.S. and will bring her young child with her. They both will be traveling with B-2 tourist visas. The notarized permission from the child's father is required to transport the minor out of the country. However, let's say hypothetically that after a few months together, the couple decide to just stay together and get married.

Question:

Would the woman in this scenario need an updated permission from the child's father to file for their Adjustment of Status, and would she need his signature on any documents?

I see one red flag and it is not about updating permission for the child to stay longer. B-2 visa is only for 6 months and that is at the discretion of the officer admitting the party to the US. He can, upon having certain questions answered, not allow the entire six months. So basically 'the woman' would be illegally staying over the limit if she so decided to stay and get married. So she really has a lot more to consider at this point.

I am assuming from your other posts over the last few months that you are a 'sponsor' for her. So in that case, why is she not applying to immigration status but only traveling on a B-2?

Before seeking immigration status is US, she MUST HAVE sole custody of the child.