Who needs Batman, Superman can patrol both cities since they are in very very close proximity of each other.
That's a silly sentiment and not even close to being an issue. Based on his power set and the fact the he's the most powerful being in the world/universe (considered to be a God and world savior), why would any other superheroes be needed when Superman can just jet over to any city at any time to handle their business for them? Look how quickly he decided to instantly shoot over to the Day of the Dead festival in Mexico to save a girl when he briefly saw the situation unfold on TV.
Either way, the fact remains that despite all of his powers and skills, Superman can't be everywhere at once. If Metropolis is meant to be like NYC in this universe, then Superman has over 8 million people to watch over, and since the film also shows him branching out to different parts of the world to aide in disasters and rescues, I'd say he'll have his hands fill as it is.
Batman proactively fights/solves crime, does investigative work, and operates at the street level (at least in this universe so far). He's also likely doing all kinds of things and stopping crimes that the people in Gotham don't even know about. I doubt Supes would be jetting over to Gotham every night to do Batman's job for him, unless there was a larger threat or incident that required his help.
Also, there's a certain level of movie/superhero logic in shared universes that must be applied here, otherwise these pointless questions (or "criticisms") will always come up. Why don't the rest of the Avengers show up to help out with the huge incidents and world-ending events at the end of most non-MCU movies? Why wouldn't Superman show up to help Batman and patrol Gotham if their cities are close to one another? Why don't the most powerful heroes always show up in the comics to help out their buddies who are facing the most serioues of threats alone?
Because while they exist in a shared universe, these writers/artists/directors/studios also want to tell separate stories featuring their individual heroes...and these heroes supposedly have plenty of their own business to take care of which keeps them busy most of the time, so they don't need all need to be used as a crutch for every story they're trying to tell.