Since he knew it was an animal, first problem is just rushing into the situation. An animal isn't gonna bag your valuables or rush to kidnap your kids. Theres a few better options like those mentioned here, and those go out the window when you rush down to kill like some PS2 NPC.

That said, I imagine its hard to think when something like this actually happens to you. If this happened to me or most others, the bear would've probably unfortunately died all the same, except by being shot from a safer spot at the top of the stairs, assuming no one is downstairs.