I like this idea, though I think that Dick would be more likely to be "global Batman" or "JLA-Batman," as his former duties (and memberships in the Titans and whatnot) have shown him taking on a more globe-spanning role, while Bruce has maintained for years that things like the League take him away from the time he'd rather be spending in Gotham.
Good point. I'm not really that familiar with Dick/Nightwing's role in the comics.
It's just that it seems logical for Bruce to be the "senior" Batman. And he has been part of the JLA -- and the Trinity of heroes at its core -- for a very long time. Besides, if Bruce knew that someone he really trusted was bearing the mantle of the Batman, he might conclude that Gotham was in safe enough hands for him to step away from it. Indeed, one could make a case that Dick, being saner and less angsty than Bruce, might even help make Gotham a brighter place than Bruce could as Batman. And hanging around more with Clark and Diana would be good for Bruce's sanity.