I have an Axiom aluminum rack that had a piece of metal inside and I squirted a few drops of krazy glue into a welding vent hole to stick it in place. It's been silent since. I might be inclined to try something similar here, but maybe with a different type of glue. Something runny so you can distribute it around the inside of the wheel without having to use too much. And tacky so it'll grab the loose piece as you shake/move the wheel around. Maybe squirt something like 3M Super 77 into the valve stem hole, then spin the wheel to distribute it and see if you can get the loose piece to stick to it.