I don't think there is any need for him to take her to court about not being listed as the father. Even if he has learned, or suspected it was MT, going to court doesn't change anything.
They can't bring Pixie back, him being on the birth certificate doesn't change the outcome. He knows he was the father, a piece of paper tying him and Sophie together forever is probably not helpful to him at this point. There is no child support or parenting payment he would be entitled too. The amount he'd be able to claim from the govt is minimal and he owns his business so he'd have to not work for that time.
The fact that she didn't let see him Pixie is heartbreaking, but going to court won't change that.
He is better off moving on and not dragging things out and slowing his own grief recovery.
And honestly - Sophie would thrive if he was taking her to court. Any attention is good attention for a narcissist. Him quietly going about his life, without her, is far more affecting to her.