Harvey's other self is part of his unconscious mind. What else is part of the unconscious? The id, responsible for things like libido and survival instincts. With that in mind, the things that the other Harvey says and does makes a lot of sense.
Fridge Logic
In "A Chill in Gotham", the kids don't wear any masks when they attack the drug dealing operation in the gym, and their faces have been on the news. So now the drug dealers should be able to tell the police who the Gotham Knights really are.
Bear in mind, fights are really chaotic. Adrenaline kicks in, it's a blur, your brain goes into fight-or-flight, and anything not immediately relevant gets discarded, unless you have lots of experience. Additionally, Stephanie has probably never been on TV before, Harper and Cullen's faces probably haven't been seen on TV in a few weeks, and when Turner shows up in photos and TV, most likely he's wearing a suit and tie in a gala and standing next to his dad, not black tactical gear storming a drug dealing operation (it's the Clark Kent effect—you don't expect to see someone in a particular location, so they're harder to recognize).
Fridge Horror
As of Episode 2, all of Batman's stuff in the Batcave is now in the hands of the GCPD. While in one sense this is great in that some of it might help them fight crime, the GCPD is one of the most infamously corrupt organizations in any DC medium. How much of Bruce's tech is going to end up being sold on the black market to the very criminals he used it against?
Similarly, it's possible that Batman's journals, containing who knows how much knowledge, are now in the hands of the League of Assassins.
Apparently, Gotham's social services are so meager that the city is fine with letting a child grow up in Arkham Asylum untended except for her mother, who's an inmate, and the occasional kindness from a warden. That's bad enough. But what really tips it into fridge horror is wondering how many thousands of other children in Gotham have had to grow up in terrible circumstances, because it seems like the system is practically nonexistent there.
Plus, if child welfare is so neglected, how many other adults have been failed by social services as well?
