For the sake of Google-searchers who are similarly mystified, I'll post the resolution.
With your BR-6600, pull the pad off the calipers. Then screw the fixing bolt back into the fixing nut, just a turn or two, not all the way. Then wiggle. Then tighten it all the way. Then try to wiggle, and it won't.
That's right... when it's loose, the fixing bolt is designed with play in it, so you can adjust toe-in and other things. Once you tighten it back down, it's fixed in the position it held while being tightened.