Is it permissible for a Muslim with a knee injury to accept a ligament transplant from a "tissue bank" belonging to non-Muslims, given the existence of other alternatives that the doctor refuses to use?

If no other treatment is available for your knee, then there is no harm in replacing the ligament with one from a non-Muslim. It does not matter that his body may have grown from unlawful sources, because your body will not be nourished by it, nor will it remain in you forever.