The State is unwilling to let any potential responsible male off the hook for child support. I doubt even a document attesting paternity at the hospital will outweigh the State interest, so I believe Dutch is right after all. It might be difficult, on the same premise, to remove the legal husband from the list of potential financial responsibility even for a child born within some months after the divorce is final.
I worked in a paternity testing lab once for a while, but I was doing research on antigens, and I didn't pay much attention. The girls at the front desk, and the girls who did the paternity tests, sometimes gossiped about how many potential fathers some babies had. The State wouldn't let up on any of them as long as it couldn't be absolutely ruled out.
Here is a bit from Wikipedia on the point:
https://en.wikipedia.org/wiki/Paternity_(law)