CTL, you’re proposed solution is theoretically good. The problem is that in a place like New York City or Israel it is not rare, as you assume. In fact, quite the contrary, it is quite common unfortunately. There are r”l very frequent utilization (virtually daily occurrences in municipalities with a significant Jewish population) of going to arkaos kneged halacha. And Jewish judges in these places will often get these cases. And the reality is they do not recuse themselves from every one of them.
Also, cases these days are generally randomly assigned to judges by the courts. The judges do not get to select cases they’d like to handle.
Furthermore, a Catholic judge would have much easier time avoiding Catholic divorce cases than a Jewish judge would have avoiding any case involving Jews that don’t have a heter arkaos. The Catholics are less limited in what is religiously objectionable for adjudicating in a secular court than Jews are limited by virtue of the fact that the default in Jewish law is that using non-Jewish courts is prohibited barring extenuating circumstances recognized by halacha.