Joseph……..
Only a theoretical answer can be given to a hypothetical question.
Until an actual occurrence occurs we’ll never know the action of a particular judge.
As for NY being a busy place with many lawsuits by Jews, NY is a big place and as such has many more judges than smaller places. That said, a judge can only handle so many cases per week and NY has more judges. The administrative judge (and his clerk) assigns cases. The indvidual clerk and judge review assignments and it is common to refuse, trade off or otherwise duck cases. This is far different from recusing oneself from a case. In that case the litigants and or their attorneys appear before the judge and the judge explains that because of some bias, connection, prior knowledge, personal connection, etc. he/she cannot hear the case fairly and withdraws.
I don’t live in NYC and I certainly am not running for a position as a civil court judge, but if I were and wanted to ease the possibility of adjudicating these cases that ‘should’ have been brought to a beis din, I’d ask the administrative judge to assign me to a court sitting in less Jewish populated area such as Richmond or the Bronx.
BUT, this is all an exercise in Hypothetical and Theory, neither of us are the candidate and it is for him to decide what to do should he be elected and sworn in.
Lastly, “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.” is something the litigant(s) has/have chosen to ignore if suit was brought in civil court. The State cannot make people adhere to any system of non-civil law (Halacha, Canon, Sharia) if as residents or citizens they choose to make use of the civil courts they cannot be denied.