Is it permissible in Islamic law to demand financial compensation for damages resulting from the inability to use a car due to the dealership's refusal to hand over the necessary documents, or does this not justify resorting to a court that rules according to man-made laws?

First: If the car was used, there is no right to claim compensation; only the right to demand its documents. However, if it was not used due to the inability to license it, the dealership can be demanded to pay rent for the car from the time of the court ruling, because it is considered a usurper, and a usurper is obligated to pay rent for the usurped item in exchange for preventing its benefits.

Second: It is permissible to resort to man-made courts to obtain a right if no other means are available, provided that only the right is taken and resorting to them is disliked.