Answer

There is, of course, a special sensitivity when the court is Jewish, since that is a more blatant raising of a hand against the Torah of Moses.
But aside from a small and insignificant minority, even today an overwhelming majority of halakhic decisors would permit going to civil court if you receive permission from a religious court. But in everyone’s view, it is preferable to litigate before a rabbinical court, and only if that is impossible should one ask permission to litigate in secular courts.
Indeed, there are various statements regarding those who participate in the legal system—judges and lawyers—even to the point of not counting them toward a prayer quorum. The basis of these statements is that even if there is permission to litigate in secular courts when there is no alternative, that does not mean you are permitted to take part in this system, which raises a hand against the Torah of Moses (especially when the system is Jewish).
Seemingly, that is the required conclusion, and the wonder is why the other halakhic decisors, with their thundering silence, express a different position. And my answer is that even if none of them dares say it out loud, they actually understand that there is no real option to conduct ourselves according to rabbinical courts, for several reasons: 1. The legal method is outdated and problematic. 2. There are no enforcement mechanisms and no authority to summon witnesses and demand evidence, etc. 3. Because even today religious courts do not judge according to Torah law, but rather make a compromise (close to the law). Before the litigation, they sign an arbitration agreement that cancels the possibility of appeal and removes from the religious court the responsibility to judge according to Torah law in its full truth.
Therefore I have written several times that, in my opinion, this embarrassing situation stems from a lack of honesty. Ostensibly there is a dispute between the school of thought (which mainly includes religious judges and jurists) that holds there is no prohibition of secular courts in the courts of the State of Israel (“courts in Syria”), and the school of thought (almost all the halakhic decisors, Haredi and non-Haredi alike) that holds these are secular courts and permission from a religious court is required in order to litigate before them. And I claim that neither these nor those are the words of the living God. The truth is that these are, of course, secular courts, and nevertheless one must litigate before them and even participate actively in the functioning of the system. The reason is that we have no real option to run an effective legal system according to Torah law (both for the reasons above, and also because of the non-agreement of secular people). And after all, our society is mixed, and there must be an effective legal system to govern the relations among all of us. Therefore one should be honest and permit litigation before them even without permission, while at the same time being honest and saying that this does involve the prohibition of secular courts, except that there is no alternative.
I wrote about this, among other places, in this article of mine:
https://mikyab.net/%D7%9B%D7%AA%D7%91%D7%99%D7%9D/%D7%9E%D7%90%D7%9E%D7%A8%D7%99%D7%9D/%D7%A2%D7%9C-%D7%90%D7%95%D7%A8%D7%AA%D7%95%D7%93%D7%95%D7%9B%D7%A1%D7%99%D7%94-%D7%9E%D7%95%D7%93%D7%A8%D7%A0%D7%99%D7%AA-%D7%A7%D7%A8%D7%99%D7%A6%D7%95%D7%AA-%D7%95%D7%A9%D7%99%D7%9E%D7%95%D7%A9