Purged files or not, Gordon isn't stupid. And what Gordon knows, Batman will know. If Jerome did become the Joker, bye bye to his being a mystery. Gordon will know EXACTLY who the Joker is...or was.
That made me think of something else. As close as this version of Gordon and Bruce seem destined to become, I can't see James not figuring out pretty quick who it is in the gray and black costume, terrorizing criminals and doing a pretty good job of handing people over to the police.
How will Gordon NOT know that Bruce is Batman?
Of course, pre-Crisis (if memory serves) he even figured out his daughter was Batgirl, and at least at first never let on. I suppose the Gotham version could strongly suspect Bruce, and just not let on.
Back in Frank Miller's Batman Year One, Gordon did the math pretty quick after interviewing Bruce as a Batman suspect. Bruce helped save his baby though so that earned Gordon's gratitude and silence.
Batman Dark Knight Returns, Gordon definitely knew and made a slight reference to Year One.
In Batman TAS: Gordon never seemed to know. He may have suspected but he never knew.