I'm not sure whether he can sue for custody. The article says he gave up all parental rights. Being forced to pay child support might be a game changer, though.
Giving up parental RIGHTS does not mean that parental RESPONSIBILITIES are also relinquished.
I used to have a friend that was going to sign over his rights cause he could not afford his child support. His mom started making his payments so he would not have to do that. IDK about in other states, but apparently in TN, it does
It's the same in Indiana...a close friend of mine signed over his rights to his son years ago there...he then did not have to pay child support after that.