Gordon doesn't strike me as the kind of person who follows or even cares about what Bruce Wayne does. He probably doesn't even know Bruce Wayne and Rachel Dawes were childhood friends, so I don't know how he'd make any connection there.
I'll agree with that. Gordon is not the type to care much about billionaire playboys (teenage girls maybe, but not an overworked police officer).
Even after the crash Bruce was doing a great job playing dumb "You think I should go to a hospital?" "You don't watch the news much do you Mr. Wayne?"
Of course I'm sure Gordon knows that Batman must be rich (or at least work for/with a rich person), but I'm sure there are at the very least tens of thousands of Gotham citizens rich enough to fund Batman.