It's sort of contradictory because they act like Beyond Birthday is his real name, but it also says that he named himself that. If you think about it, that can't be his real name because not even Wammy's orphanage had information on his name, just like Near and Mello. Beyond Birthday was the name he was living under in LA.
But you're also right, that Kira couldn't have killed him if Beyond Birthday wasn't his real name. Unless Misa or Mikami saw his picture, which is not likely since he was probably pretty mutilated after the events of the book.