Is it permissible for a doctor to deceive or lie, verbally or in writing, to facilitate treatment and examinations for severely impoverished patients who cannot afford to pay, given that the hospital director refuses to approve their treatment while approving free treatment for individuals of prestigious status?

The default is that it is not permissible to resort to trickery and mislead officials, and lying is forbidden except out of necessity or a need where no other legitimate means exists to achieve the purpose. The criterion for permissible lying is: every praiseworthy matter that cannot be achieved except through lying. So, if you cannot restore what is rightfully due except by lying, then it is permissible for you to do so, and you may be rewarded for it if your intention is sincere for Allah's sake.