Humanely killing of animals can also be the moral choice. When over population is starving or sickening an animal population, hunting can absolutely be the humane solution.
Not when it's unnecessary.
A lot of people think that hunting can be "moral" when there's a supposed overpopulation of certain animals, but what many people don't realize is that in many cases an animal population is artificially boosted to increase recreational hunting opportunities for hunters.
Furthermore, even if that wasn't the case, killing is not the answer. There are other ways to deal with certain problems that don't involve killing.
We (mankind) were given one job. To rule and take care of the animals as representatives of God, which means according to GOD's perfect will, not our own. God's perfect will is crystal clear both in Genesis 1 and in the prophetic scriptures about the future Kingdom, when God restores that initial paradise that was HIS intent for creation in the first place. Peace and harmony, not exploitation, cruelty, needless violence and killing.
We have not only failed at that one job we were given, but we have done pretty much the exact opposite, to the point of being humanity's greatest shame, imo.
Certainly carnivores and omnivores will kill for food and other than humans, none give a flying fig about doing it humanely and most don't. I know it happens but I refuse to watch it live or videos of it happening. That's not cognitive dissonance though. It is a personal choice not to unnecessarily undergo distress over something that is entirely natural and happens whether or not I can stand to watch it. I do not feel it would be right to prevent it from happening.
What animals do should not be an excuse for our behavior. Some animals also rape and eat poop, that doesn't make it right for us. Also, carnivores like lions don't have a choice. We do. Meat is not necessary for humans. In fact, as was stated before, a diet high in animal flesh is linked to cancer, heart disease and other preventable illnesses. We were never meant to eat corpses, to eat death and violence.
But would I shoot the animal that is attacking my dog or another human without giving it a second thought? Absolutely. The coyote going after my newborn lambs or calves. Yep. Again choosing the most moral choice between two bad things.
That's a different matter. As someone already said, self-defense is not the same thing as murder or needless killing. Self-defense is justified. I don't believe that needlessly killing a sentient being who wants to live just as much as you and I do is ethically justified.