What is the ruling on working in the legal department of a commercial institution, which includes litigation before courts and following up on insurance claims, knowing that the applicable law is man-made?

If your task is to follow up on files and matters related to the internal laws of the institution, and these are not contrary to Sharia, and are merely administrative tasks, then there is no harm in carrying them out. However, if they involve injustice or an infringement on the rights of others, then it is not permissible to follow up on them.

As for litigating before man-made courts to establish or deny the rights of others, if you find an alternative, that would be safer for your religion and more cautious for you. But if you do not find an alternative, there is no harm in adopting the view of those who permit it in circumstances like those of your country, however, it is not permissible to defend what is false.