Here's how I would have done The Batman to fit in the DCEU
Bruce Wayne is a billionaire who's parents were killed when he was young and returns to Gotham after a five-year hiatus where he trained with the League of Shadows. Oswald Cobblepot, aka Penguin runs a illegal club that caters to corrupt officials, like DA Harvey Dent. He uses this and his Red Triangle Mob to curry political favors and run for mayor. When Wayne learns his company's funding the campaign, he fights the corruption as Batman. Shortly after, the mysterious Riddler starts killing corrupt officials and kidnaps Dent, who had his brother killed. Chief Jim Gordon & forensic scientist Patrick Parker work with Batman to decipher Riddler's clues. Batman tries to question Penguin but his Gang saves him, so Batman kidnaps member Dick Grayson and trains him as Robin for info. Riddler burns Dent's face half off on and calls him "Two-Faced" on TV. Batman cracks his code and saves Dent but realizes Dent's going to kill corrupt officials at his private gala for abandoning him. Dent flips a coin and kills Penguin but Batman & Robin stop him before more deaths occur.End credits: Luthor meets with Patrick where they turn Kryptonite red.

Cast
Bruce Wayne - Ben AffleckAlfred - Jeremy IronsDick Grayson - Robert PattinsonHarvey Dent - Guy PearcePenguin - Colin FarrellPatrick Parker/Riddler - Tom Hardy