Sounds like his identity is public now? Wouldn't that give Turner Hayes (Bruce's other, other, other, other adopted son) some trouble, seeing as how he's likely to be a vigilante in the show?
Harvey Dent is apparently not Two-Face in S1. With the Arrowverse, this would work fine because Harvey was only referenced as Two-Face in an alternate future. If you add Gotham into the mix... the Pennyworth tie-in mentions Two-Face among other Batman villains, meaning the show would have to take place before the comic. That would cause some issues with the 40 years thing, and Bruce is mentioned as being alive too. (Maybe Alfred is the only one who knows Bruce is alive)
Of course, Gotham Knights is a completely new continuity as far as we know.