Joseph….
In your hypothetical the Frum judge should simply have his clerk remove the case from his calendar for reassignment to another judge. He need not publicly (or privately comment as to why). The chief administrative judge would not likely ask why for these very rare instances.
Do I think the Frum judge would do so, probably…see next paragraph.
In the 1980s I remember an Irish Catholic judge who served in Superior Court (trial level court) in Connecticut. Judge F was very observant in his faith and did not believe in divorce as an option for Catholics. Every few years he would get stuck with a 3 month term in Family court. He’d attend the scheduling meetings and put his name down for custody and alimony cases and divorces that didn’t show the parties to be Catholic (done by examining the marriage licenses in the case files, ceremony not conducted by or signed by a priest). He’d pass on divorces between known Catholics letting Jewish or Protestant judges take the assignments.