Very good point.Benzodiazepines?
Benzo overdoses result in confusion, drowsiness, lack of coordination/balance, lightheadedness, muscle weakness, memory loss, fainting, slowed heartrate, difficulty breathing, and shallow breaths. Death is the result of paralysis of the diaphragm or heart.
These symptoms seem like the animal would (hopefully) be unconscious by the time of death, and it doesn't seem to result in any particularly painful symptoms. Unfortunately I'm not sure what the side effects may be in non-human animals like birds.