What you're describing doesn't really [i]need[/i] to have an old relationship with Batman though, or even a past as a crime fighter. There are plenty of people that dislike the idea of the people rallying behind a single costumed vigilante and believe in the strength/power of the people. Hell, Lex Luthor is one of them, albeit an evil version. There's also detective Dan Turpin. And I just realized that I listed two Superman characters, but the point is still valid. :P I'm not personally against using Jason in this way, I'm just saying it'd be easy as Hell to make an OC fit the mold you're sculpting. 

Edit: Re: Dick being a cop. Well he's a cop in Bludhaven, so you know. No proper jurisdiction in Gotham. :P