You're both.

[ What is he supposed to do with this? Damian is unprepared for dealing with well-adjusted children beyond swooping in, saving them, and dropping them off in the arms of their guardians in disgust.

He's mildly tempted to toss this boy into the water and make a quick getaway. ]

The immortality is a little better. The average species of Papilionoidea lives less than two weeks. Not a long-lasting pet.

[ Immortality for Titus and Alfred the Cat, now that's a thought. Perhaps Zatanna might be able to--

The thought, as quick as it comes, is dismissed. It's foolish. He's dead, and his father would never compromise his own ideals in using the Lazarus Pit to bring him back. No, unless Damian can somehow claw himself back into the land of the living from whatever afterlife he will find himself in (or has found himself in), there's no going back.

The longer he thinks on this, the sicker he feels. ]

It'll come, in time. [ Best not to think on it at all, for the moment. ] No need to rush it. I hear childhood is something to be enjoyed.