Are you thinking of the “Butch the Steer” story?
@Momof3teens - we got a bottle calf when my boys were 10, 8 and 5. They named him Butch because, “We’re gonna butcher him, anyway, Mom.”
You are the parent. It shouldn’t be up to the kids to “let” you do what’s best for an animal that’s not thriving. It may be time to sit down for a matter of fact conversation with said child and explain that sometimes the kindest thing to do for an animal is put it down. Making hard decisions is part of animal keeping. It’s our responsibility to not let them suffer. I’m not saying the kid should be present when it happens if they’re going to be traumatized by it, but it is a fact of life they are going to have to learn to deal with.