Aside from The Question, Batman's the closest thing the mainstream DCU has to Rorschach. He doesn't, and shouldn't, compromise.
Considering that he still refuses to kill any or all of Gotham's repeat offenders--even after they've murdered thousands and thousands of people--simply to maintain his sense of ethos and adhere to the demands of his own Egoist principles, I'd say he's definitely a believer in Rorschach's Armageddon.
Quite frankly, there's no way he could pull the trigger. He'd be physically and mentally incapable of using a gun to kill someone--let alone murder his own parents.
