If you're punching the hole to the same size as the eyelet, it may be to tight to turn. Try punching the hole slightly bigger, place the eyelet, and don't overset. The bottom page would stay stationary and only the top 2 sheets will move ( or visa versa if you set it the other way)
Also, there's another type of eyelet that doesn't need a setter, you get two pieces and one "snaps" into the other. This may be easier for what you're trying to do, but I've only seen these in plastic.