I think to kill for the sake of killing, is totally wrong
To kill for food, in a humane way is ammicable, to kill just because it annoys you is ridiculous, considering we have annoyed the mockingbird building around the trees.
I suggest you get a cat or do something that removes the problem over a few weeks, you shall find it vacates the tree & moves to another, lol, maybe in a tree in your back garden
