Solution: Dispatch the rabbit using cervical dislocation (breaking the neck).
Explanation: This method, often called the 'broomstick method' (using a bar/stick behind the head and pulling sharply upwards on the hind legs) or by hand (holding the head and hind legs and pulling/twisting sharply), causes rapid unconsciousness and death by separating the spinal cord.
Notes: Requires practice to perform correctly and quickly. Considered humane when done properly. No special equipment needed besides possibly a metal bar/pipe.

Humane Dispatch: Cervical Dislocation