What is the evidence for the permissibility of resorting to man-made courts when necessary, with the intention of ruling by what conforms to Sharia, and what is the extent of validity of the statement that this is an act of disbelief?

The general principle is that a Muslim should only seek judgment from what Allah Almighty has revealed. However, in the absence of Islamic courts and when a person needs to resort to man-made courts to repel injustice or establish rights, there is no harm in doing so, in order to achieve certain benefits or avert certain harms, and in application of the rule of necessity, to secure rights, and to reclaim what is due when there is no legitimate alternative. Furthermore, not all provisions of man-made law contravene the rulings of Islamic Sharia. This is supported by the statement of Sheikh Ibn Uthaymeen, may Allah have mercy on him, who permitted seeking judgment from those who rule by man-made law to preserve rights that might otherwise be lost, for truth is accepted from anyone.