Solution: Use a high-powered pellet gun (air rifle) aimed at the brain or a captive bolt gun.
Explanation: A precise shot to the brain causes immediate unconsciousness and death. A captive bolt gun delivers a forceful stun, usually followed by bleeding.
Notes: Requires accuracy with the pellet gun. Captive bolt guns are specialized equipment. May be preferred by those uncomfortable with physical methods.

Humane Dispatch: Pellet Gun / Captive Bolt