Me myself I think that killing an animal isn't cruel in itself. That is the natural order of things.
HOW you treat an animal is a different kettle of fish. Which is why it hurt me so much to watch Food Inc for example. Seeing how those animals were treated falls under cruel to me.
Though many vegetarians will disagree with that point. Many think that because we are humans we are not omnivorous animals that are subject to natural law... that we have a choice and because we have the brain to make that choice we are morally obligated to do so. I understand that view, and respect a vegetarians right to their view. In my personal experience most vegetarians are just as respectful of my view, even though they do not agree with it.
But, I also hate to see waste. So killing an animal just to let it rot... that's just not the natural order of things. Animals, including us, die and decay yes. But if you kill something it should be to protect yourself or to sustain yourself. Nothing wrong with a dog killing a rabbit and eating it because there isn't any other food. But a dog killing a bunch of chickens just for sport, that's waste.
One mentioned that she knew folks that sacrificed a chicken and then once the spirits had taken what they needed from it the family then ate the bird. Nothing, even spirit according to them, is wasted. That, to me, is no different than me killing a chicken, skipping the spirit bit, and cooking it up.
That's my two cents on that subject. Donno how it applies to the OP's stuff. You mentioned that they cook the bird after some rituals, and the word cook to me indicates that it's being prepared to be eaten... so that works for me even if I don't believe in it. But you didn't specify for sure so I won't comment one way or the other.
But, in the end, I'm not responsible for your actions... only my own. I have to live with my choices as does everyone else. So basically as long as you aren't hurting me and mine it's really not my primary concern.
Though it does help to understand the feelings, and motivations, of those around us even if we cannot totally empathize.