You're a genius, Doug. I rolled it to the thickness of card stock. Let it dry for a while (probably could have dried longer, but I was impatient). It worked great! I left the bottom layer of paper on it for final drying, so it was easy to handle, even though it was so thin. I'm planning to use wafer paper as a backing sheet when doing this to apply directly to cakes. That way, I can leave the paper on and just slap it on the cake. It'll hold its crisp lines that way and not get bent out of shape.
The machine is expensive, so I don't know if I'd encourage anybody to go out and invest in this. But, if you've got a machine already, it's neat to know you've got this in your bag of tricks. Some of the layered cutouts can be really nifty.
I'm attaching a picture of my last tag. It's probably a shade under 3 inches long. I'm planning to write birthday greetings, etc., on this with edible marker and then attach it with a thin strip of fondant, probably under a bow.