The "real" Batman would be the commonly mainstream current incarnation of Batman.
If Batman were a killer now then he would be the "Real" Batman, but he's not a killer now because that's not who this character is. Having this character kill would be out of character and it would change the nature of his identity. He would turn into the "Lawful Evil" Batman of DC Comics 27-34 (ish) or the more Brutal Frank Miller or Tim Burton Batman.
You have to realize that many times you cannot just change the nature of an accepted character for the purpose of "it would be neat" that cheapens the development that we've had with Batman.
If Batman killed then he would simply become a killer and as Green Arrow says, "We're heroes. That's not who we are."
And yet Oliver Queen started killing bad guys in Mike Grell's series in the late 80s/early 90s (which is still part of his canon).
If Grant Morrison's statement of adding ALL of Batman's comic adventures to the modern day canon is correct (and his stories for BATMAN back that up) then retroactively he has killed.
Prior to that Batman does kill an alien in COSMIC ODYSSEY (1988) with a laser gun and in the BATMAN/SUPERMAN story Enemies Among Us is prepared to use a nuclear weapon in order to stop an alien invasion. I think if they put the character in certain situations (like those) killing IS permissable.