GH:
The matter at hand is not the recognition of a psak beis din al pi halacha. The beis din does not have legal status as a court, and that is ancient history. The matter here is the role of the beis din as an arbitration panel. There are many other forms of arbitration besides beis din. And there is no reason that BD cannot serve as one, even though there is a religious aspect to it. Various states may have laws that limit the authority of an arbitration panel. Just as relevant is that there are often agreements that are not the result of “litigation” by either a court or BD. If two people agree on something, they should be held to it. These agreements, often mediated by toanim or other negotiators, are filed in court, and become legally binding. Can these agreements include commitments in how to raise the children regarding religion and lifestyle? From the report in the OP, it seems that a judge is stating that this cannot be done. I question that (I am not a lawyer). I would not be surprised to see another court overturn this ruling on appeal.
A religious court has no standing as a court in American law. But it does have standing as an arbitration panel. And agreements are not arbitration outcomes. They are agreements that the two sides reached, that were not imposed on them. I fail to see how a court can block that. Waiting for round two.