In my opinion,the answer is NO. and here's why:
In terms of being Batman, his purpose is to BE mysterious. So Batman should never reveal his identity anyone other than those who already know. This would really foster the working relationship.
If for some reason (as in BTAS : "The death of Batman/Batgirl", if Gordon should find out that Bruce is Batman, or should the police or whoever find out, then the character must be so distinguuished that no matter what, when it comes to crime or serious business, they see both Bruce Wayne & Batman as 'Batman'.
They should only 'see' Bruce Wayne or Batman as 'Bruce Wayne' in social functions or in private moments, say when Gordon or Harvey Bullock are hinting or mentioning (the conversation shouldn't be long like a chit chat) about some personal need or grievance or opinion looking upon him as a trust worthy confidant.
On the outset I think the question, is how on earth to distinguish Bruce from Bats whether or not in costume and how the police are going to keep quiet knowing the identity.
For serious Business, Batman must build his reputation, firmly affirm his attitude and never reveal his identity. This way, the Batman persona will stick and the vibe will announce to those who know, which persona they are interacting with.
As Bruce Wayne, he is seen as light hearted as normal and fun civilian playboy. People who know the dual identities and the working styles of both Bruce Wayne and Batman, would obviously be clear about his work attitude and its effectiveness.
Hence, they will also subconsciously distinguish Batman and Bruce wayne based on the situation, his attitude. This along with his not consciously revealing that he is Batman, will add to the total mystery that's:
As Batman in costume: "The powerful urban legend"
As Batman out of costume: "He is the Batman...he will know what to do,
but I'd rather approach Batman, than the civilian for help" or "Even as Bruce Wayne, he IS the Batman...he'll find a way to do all that Batman can, even without his costume or gadgets."
As Bruce Wayne in costume: "He's the Batman, the real deal, but I believe he'll understand what I am mentioning to him,...hey! and again, he's gone"
As Bruce Wayne out of Costume:"Ha, here comes the indutrialist, we can joke about the latest starlet on broadway".
To build this clear projection of attitude, he needs to establish his attitude and work effectiveness as Batman and Bruce Wayne seperately, to the point that nobody can accept that Batman and Bruce Wayne are the same person, even if they are out of costume and its physically obvious. Till this point, even the danger of being revealed should be taken care of.
