Jews, male or female, should not be taking a position as a judge on a non-Jewish court that will have cases involving two Jewish parties litigating against each other. It is forbidden for Jews to litigate a case against each other in non-Jewish courts. And it is forbidden for a Jew to judge a case between two Jews who are non-halachicly using non-Jewish law instead of halacha (especially if one of the litigants doesn’t want to be in non-Jewish court or using non-Jewish law, but it only there because the non-Jewish law and authorities force him to attend and respond to the case.)
Secondly, women shouldn’t be taking public positions.
Sadly, whenever this is pointed out (about becoming a judge on a non-Jewish court being halachicly wrong), despite explaining the extremely severe issur arkaos involved, the retort of supporters of Jews becoming judges on non-Jewish courts invariably is that “he asked a shaila”. (Unfailingly to a rabbi who must remain anonymous too.) They virtually never can offer a halachic rational explaining how they can overcome the serious prohibition against arkaos and, specifically, judging cases between Jews who are in non-Jewish court Keneged Halacha (at least one of the parties to the case, with the other Jewish party simply being dragged into non-Jewish court against his will.)
Well, at least they can say “he’s wrong” and leave it at that.