There are specific requirements for custody arbitration to be legally binding. This is from Fawzy v. Fawzy:
Finally, the Court addressed the process by which parents can bind themselves to arbitration over a child custody dispute. The Court emphasized the importance of ensuring that parents understand their rights and the ramifications of their decision before they agree to arbitrate. The Court thus required there be an agreement in writing or otherwise recorded that clearly establishes the parties 1) understand they have a right to a judicial determination of their dispute and knowingly agree to waive that right; 2) understand and accept the limited circumstances in which they can challenge the award; 3) have had enough time to fully consider the consequences of their decision to arbitrate; and 4) have entered the agreement voluntarily.
If a Beis Din stated that they wouldn’t preside over a Get unless she agreed to custody arbitration, than such an arbitration can be challenged.