Does filing a lawsuit to claim unpaid financial dues from a company that withheld them fall under seeking judgment from Taghut, and is the money resulting from that considered a necessity?

If the contract between you and your employer violates general labor law, and your employer has not wronged you in anything you agreed upon, then you are not permitted to demand anything from him based on labor law. This is because labor law, in this instance, is akin to a custom that holds no weight when explicitly contradicted. The mere fact that a court might compel him does not justify wrongfully consuming his wealth. However, if the contract between you was in accordance with labor law and did not stipulate anything that contradicts it, then you are permitted to demand from him what the law stipulates, as long as it does not contradict Islamic law. The fact that courts are man-made (positive law) does not prevent resorting to them out of necessity to reclaim one's rights and remove injustice, and this does not constitute shirk (associating partners with God).