Yes, Batman's costume, voice, and methods are very theatrical, but what is truly Batman is his personality and his drive.
Bruce is a cover because he is shallow, loves to party, frivolous with his money, cowardly, a disgrace to his family's name.
Batman is driven, focused, brave, fearless, couragous, true, and all that good stuff.
In Batman Begins Bruce pretends to become sloppy drunk and insults all his guests, including many friends of his parents. He lets people think he burned down his own mansion. He pretended to date two girls at the same time. Bought a restaurant to show how rich he was, and to look sort of snobby and mean. He acted like he had no care for business.
In Dark Knight Bruce slept through an important business meeting, looking unconcerned about his own company. He crashed his car to save a guy's life but then pretended not to even know he saved anyone, he was merely driving recklessly.
The Batman is a symbol he created, but so is Bruce. His true person is a Batman that looks and sounds like Bruce.