1/4" plywood is actually not that difficult to cut. How about a "stamping die"? You could get one in a donut shape. You would need a press, probably something that could exert a couple of hundred pounds (or more) of pressure.
I've used stamping dies to cut plastic sheets (1/4-1" thick). I've used knives to cut 1/4" plywood. I don't really see any reason why a die couldn't cut plywood.