Is the wife's waiver of her right to custody before a non-Muslim judge binding on her, and is this judge's decision regarding the children's maintenance (nafaqah) binding on the father, given the absence of an Islamic court or an Islamic center for dispute resolution in the Western country of residence?

If the spouses agree to resort to an Islamic dawah center for arbitration, then they must do so. The mother's relinquishment of her custody does not require a judge's ruling; her acknowledgment is sufficient. As for maintenance, the father is obligated by Sharia (definition) to provide for his children, and its amount is determined by custom. If the ruling of a non-Muslim judge aligns with custom, it must be adhered to because it conforms to custom, not because it is the judge's ruling. If there is no qualified Muslim judge, it is permissible to resort to a rational Muslim, and his judgment is binding if both parties commit to it, because "reconciliation is permissible among Muslims."