“Batman” sounds like a superhero, which works in some contexts (e.g. a Justice League story). “The Batman” sounds like the kind of monster a superstitious, cowardly lot would be afraid of, which works in other contexts (e.g. narration from the perspective of a henchman who doesn’t really no much about him). Generally, I’ll accept whichever sounds good in context, but I’m pretty sure I’ve only ever used “Batman” in conversation.
It is kind of strange that Batman is the only “something-person” superhero who regularly gets a “the.” Nobody talks about the Superman, or the Spider-Man, or the Wonder Woman. Well, actually, now that I think about it, there’s also the Elongated Man, I guess.
Final point: When you ask Bruce who/what he is, he says “I’m Batman.” I’ll take his word for it.