hmmm...I'm going to go back to the "free flight bird room" idea. I *thought* you were going to say that you couldn't keep your bike in there because the bird would take it apart(I've know some pretty destructive Cockatoos in my time; although if you have an African Grey you could probably teach him to do a complete overhaul ). Bird poop would also be a problem.

Have you considered building or buying an aviary instead?

There are some pre-built cages that are the size of a walk-in closet. I'm not sure if they would be suitable for your pet. The cool thing about the ones I've seen is that they are usually on wheels/casters and can be rolled outside (without having to worry about the cat!) on nice days.

A good friend in HS had several parrots. She built an aviary that took up ~1/2 of her room. If you did this both the bikes AND the bird could live inside.