Really? You've been there many times? Taking a road bike to Catalina makes about as much sense as taking your dog to a strip club--even if they let you, it's only going to less fun for you, and probably not very good for the dog either.
Rent a mountainbike somewhere here on the mainland (the renters they have on the island suck), then pay the $7.00 fee to take it over on the ferry, then pay the ~$35? Catalina Island Conservancy fee. A lot of trouble, but entirely worth it IMO:








Have fun!