I have only seen the metal plates (like the one pictured below, from Hunt's). They come up for auction every couple of years, and are rare.
I don't know if there was ever a full Brooklyn Dodgers Bicycle, per se. These Metal Plates were probably made to attach to a bike, simply to show support for the Dodgers. So perhaps these plates were made as ornaments/accents which could be applied to
any bike.... much like a horn, bell, reflector, tassels, etc.
Bike accessories were big back then. I know there is a Cleveland Indians Bike Reflector from the same period, and other Bike ornaments that relate directly to baseball.
Furthermore, a Google search does not turn up anything for a Brooklyn Dodgers Bicycle. So it may never have been manufactured as such...
http://www.huntauctions.com/LIVE/ima...=881&lot_qual=