He's a dick, but only because Robin lost perspective. Batman's life isn't what is important, it's the lives of everyone else; and here's Robin wasting time asking if Batman is ok when he could be out helping people who really need Robin.
He's not intentionally being mean to Robin, he just wants Robin out of Arkham and go somewhere where he is needed.
As far as the DLC goes, Batman is suffering from Trauma and is trying to ignore any emotions he might have by diving further into the Batman persona. --- "It's the stronger that make the weaker"- Milliardo Peacecraft