If you have verified your game files, the file is present, but the game still won't launch, the issue may not be a missing file, but rather compatibility issues. The bmlauncher.exe is notorious for crashing on Windows 10 and Windows 11.
A modder named created a modern replacement launcher. Search for “ArkhamLauncher 2.0” on GitHub. This tool: batman arkham asylum bmlauncher.exe download