Apparently he demanded the keys, she refused. So he shot her and took them.

If I were a criminal, I would think the best course of action would be to threaten again. If she still refuses, let her win. Flee the scene.
You live (stay out of jail/investigation) to try to rob again.

I can't see how shooting the car owner would ever be the best choice unless you are trying to uphold your reputation that your threats are credible.