One parent withholding SS Card from other parent

Location: OH, child is 16.

Parents A and B have had shared legal custody and parenting for over a decade. About a year ago, child became estranged from parent A, and has refused all contact with them since. Nothing has been altered in terms of legal custody.

It is time for the child to get a Driver's License and it turns out parent A is in possession of the social security card. They are refusing to provide the card on the grounds of "emotional boundaries", and advised parent B to take the child to the SSA and obtain a new card. A novel idea, except there is a bit of a government shutdown, and appointments are booked out over a month.

Would it be worth it to go to court and compell it, or just let sleeping dogs lie, wait the month for the appointment, and thug it out the next 2 years? I really try to avoid fights that don't need to take place; both for financial and emotional reasons. Parent A would willingly relinquish all custody, but that wasn't pursued in order to avoid emotional harm to an older child struggling with being rejected by a parent.

I'm trying to guage whether opening a 13 year old court case is worth it here. If its an easy "obviously hand over the damned card" type of judgement.