My SSR: 2004 in Black and now 2005 in Black, what a pair
Is the bracket attached to the frame welded or bolted to the frame? If it is only attached at the top then that could be the rattle at the lower section of the bracket and the frame. I would see if you can wedge something like either a small wooden wedge or a piece of rubber in there and see if it takes the rattle away.
FWIW: I had a "rattle" when I re-did my exhaust, but it only happened sometimes when I was backing up (yes, seriously). What I found was under just the right load, backing into the driveway, the exhaust cross over would juuussst touch the rear suspension brace/bolt on the driver side. It was at the top of the suspension, just before the exhaust turned to go back to the rear of the truck. The stock exhaust has a flattened section in that area too. I tried twisting and adjusting the exhaust but I couldn't quite get it. I subsequently "adjusted" that section of the exhaust with a mallet and board.