The latest two-wheeled additions to the Batman movies are the Redbird(1997) and the Batpod(2008). The Redbird is Robin’s motorcycle and it was designed by Harald Belker. The information I have found say that it was custom-built, based on a Honda engine.
The Batpod was designed by Nathan Crowly for The Dark knight. It has crazy 20 inch tyres and a single cylinder engine is mounted in the hub of the wheels.