Is it permissible to grant a clearance (bara'at al-dhimma) to someone who receives their driver's license while listening to music, when the songs might contain mockery, and would that be considered a sin?

The employee must perform his work honestly. If the car owner is entitled to a release of liability according to regulations, then he should not be prevented from it, even if he is listening to forbidden music. He should be advised and reminded of Allah, but that does not negate his right. The employee should intend by his work to fulfill people's needs.