This can of worms was opened up a hell of a long time ago, back when the laws on child support were written. "Best interest of the child" is the legal standard. Under this standard, if a woman digs up the condom that was used by her roommate and her roommate's boyfriend, and she inseminates herself with it, a judge has to find in her favor and compel that man to pay child support.
"Best interest of the child" is what allows the state to say that a sperm donor has no parental rights to his biological offspring, but that a sperm donor has financial obligations to his biological offspring. "Best interest of the child" sounds really good, proper, and noble, but it leads to this kind of stupidity when the courts are hamstrung by archaic standards of parentage.
Hell, there have been cases where boyfriends and step fathers have been forced to pay child support for someone else's kids when that someone else can't be assessed child support himself, on the basis that the child has grown an emotional bond with that man, and that child's emotions somehow obligate the man to provide care for it. After all, that's in the "best interests of the child".
When a woman decide to get a man so drunk that he doesn't even know she had sex with him (which would be a classic "rape" if the roles were reversed), our laws say that man owes support to the child born of that rape. Best interests of the child, don'tcha know.
(Of course, when two sisters do exactly this to their father, we write it down in a book and call it "religion" and use it as an example of "good moral values". But I digress...)
The really interesting part about this whole mess is that if gay marriage were broadly recognized, we'd quickly set the precedent that a person can be "divorced" from their genetic offspring in certain situations; that the legal parents of a child are defined separately from the biological parents. This precedent would eliminate a hell of a lot of problems with child support being unfairly allocated. We would also eliminate the rights of male rapists to gain visitation and/or custody of the children born of their crime. We'd be able to resolve all sorts of paradoxical parental issues simply, fairly, and effectively.
With that precedent set, the judge in this case would legally be allowed to say: "Ma'am, you and your lady friend are the child's legal parents" and the problem would be resolved.