First off, is this 'bro' your actual brother, or friend?
Either one, ask him if they could stop fighting over her with you. When two people, no matter how close to eachother, like the same person, it causes great fighting. So if you both back off a bit, your friendship could stay stronger.
Second, on what level do you like her? Who knows her better? If he knows her more, I suggest getting to know her a bit more. That way, you have even 'competition'. Is it better to know the same things about the girl with your friend/brother than not knowing much and STILL not being good friends?
If he doesn't know her as well, you should let your 'bro' get to know her. That way he can either see if she's really the one for him, or not.