Go big as possible. Much much bigger than just an umbrella.
Just think how many people do headshots in the studio with a 20"x30" or bigger softbox. Now imagine how much bigger a bike is and make your light source about that size relative to a persons head vs a 20"x30" softbox. You can do this fairly cheap, by purchasing some white sheer sheets and hang it a few feet in the front of your umbrellas. Mine is 10'x20' in the front of 3 softboxes arranged side by side.
This will create a huge softbox kind of effect and the one small reflection from your umbrella will turn into one giant uniform reflection that you can blend in photoshop later.
Here is a picture I shot with 2 30"x42" softboxes and 2 background lights.