Damian's debute featured the murder of the Spook, and Otto Netz is a character in-continuity who has cropped up recently - dead - still causing trouble, so it should still be Damian who murdered him. In both of those cases lives were threatened, so Damian is not 100% the worst human ever. But he did it incredibly callously and bruskly and business-like, so, not great, either.
At present though, he's killed three people that we know of, and they were all super-villains threatening lives, so like ... I have to imagine that in spite of striving to be better, and to live up to the Batman Code, he's not going to be Anti-Kate here.