[villagetinker]

I would check with the local building department; I believe this would be an extension of the house, and I am sure there will be strict code requirements. However, your biggest problem may be the drain line, you may not have sufficient slope for the drain (assuming your sewer connection is in the front of the house). While I realize the general rule is anything goes inside the birdcage, this may be pushing the point, you may want to check with ARC on this also. The birdcage foundation may not be suitable for your planned addition.
IMHO, I believe this will be a fairly large and complicated project. I hope this works out for you.

On a side note, if all you need is an outdoor shower, this can be accomplished much easier, I have seen a few pools with these installed inside the birdcage.