Shalom u’vracha,
Your question is important. According to halacha, it is forbidden to bring a claim before civil courts (arka’ot). This is an extremely severe prohibition, and one who violates it is considered to have raised a hand against the Torah of Moshe Rabbeinu, עליו השלום.
Even the courts in the State of Israel, whose judges are Jewish, and even in matters in which the Rabbinical Courts rule similarly to the civil courts, it is forbidden to turn to civil court. One is obligated to bring the claim exclusively before a Rabbinical Court. This is because filing a claim before arka’ot, which rule not in accordance with Torah law, grants esteem and honor to a judicial system that rules contrary to the laws of the Torah, and this is forbidden.
Only where the other party refuses to submit to Torah adjudication may one turn to civil court. However, in family-law matters, the Rabbinical Court has authority to rule just as the civil court does. Therefore, according to halacha, there is no permission to file the claim in civil court.
I do not agree with the assertion that the Rabbinical Court favors the man more. However, even if this were true, it would not justify raising a hand against the Torah of Moshe and turning to arka’ot. Hashem will send abundance and blessing from another source, and one who observes the mitzvot of the Torah does not lose out.
With abundant blessings,
Rabbi David Ohayon