I don't see how it's so unreasonable to look at TDK and take note of the regular men in the opening scene who dressed like Batman because they thought they could do his job too - which was an important part of the story. Now take a look at this movie, assume we have a "street-smart Gotham girl" who thinks she can do the same thing by also dressing up like Batman... a Bat...Girl. It doesn't have to be some fancy ridiculous Schumacher thing like you all believe would happen. It's just a girl doing the same thing the False-Batmen did in the previous movie, only she's a slightly more developed character, who thinks she can use her brains while in costume to take down Catwoman herself. 
 
Add to the fact that she's Gordon's niece, or Gordon's Whatever, and you make it a personal mission for her, since she knows that Batman and Gordon have a relationship. Perhaps she resents Batman. Or perhaps she wants to make Gordon proud. Batman would be extremely unapproving of her, just like he was with the False-Batmen. It's not like she would be hanging out in the Batcave with Bruce, making wise-cracks and doing backflips. She would be completely separate from Batman, in no way affiliated with him, doing her own thing as a vigilante. In the end, she realizes that there's only one Batman, and she stops.