As many people have pointed out Batman is not schizophrenic, that's characterised by delusions and psychosis. I don't even think he has an identiy disorder, he doesn't really have split personalities rather just layers to himself, it goes Playboy-billonaire Bruce Wayne < Batman < The hidden Bruce Wayne whose a father to his Robins, friend to Superman and see Alfred as a father figure.
If I had to suggest a mental disorder I'd think he had elements of something less severe, Schizoid personality disorder (actually unrelated to Schizophrenia despite the name). It's a personality disorder characterized by a lack of interest in social relationships, a tendency towards a solitary lifestyle, secretiveness, emotional coldness and sometimes (sexual) apathy, with a simultaneous rich, elaborate and exclusively internal fantasy world.
Nearly all of those could be applied to him at some point. You could say he's unique though that he draws people (Robin, Alfred) into his little world, but for the most part he doesn't have conventional relationships.
Another possibility is potential Post Traumatic Stress Disorder, he has elements like reliving the events and hyper-arousal, but he doesn't avoid triggers (like well gun wielding thugs) or experience any emotional numbing, it doesn't impact his chosen life that much.
Is Batman crazy? No, but he's not completely normal.
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