The chest region encompasses the area around the forelegs and the area immediately underneath where the wing connects to the body. It's still possible to be shot there. Otherwise, she was shot in the leg, rendering her immobile - The killer would then have to immediately step up in front of her with the pistol while reloading to finish the kill, which would have gotten him instantly caught. In order to be shot in the thigh, then be shot in the chest, Daring Do must have been facing either left or right from where the attacker was - hence, she must have been facing away. If they were arguing, the attacker must be someone other than Mr. Whooves, unless we assume Daring Do argues while facing the complete wrong direction.
Even if we assume that the third shot missed, the bullet to the chest was the one that killed her. That makes it impossible for any of the shots to have been the missing bullet.
It can't have been the last round, because she was already dead then.
It can't have been the second round, because the derringer reloads too slowly - the shooter would have been noticed by both Daring and everyone else.
It can't be the first round, or Daring Do would have noticed and avoided the second round.
The only shot that makes sense is that the doctor fired the gun before the incident ever occurred!
It's the only explanation that makes sense, and also explains how only one shell was found at the scene. The derringer can only hold one round. If the first round was ejected to make room for the other, only one shell would be found on the scene - The other would still be in the gun. If three rounds were fired, you would find two derringer shells on the crime scene.