I’d like to eliminate size of the piece and registration after flipping. You can’t do that easily with the stock you’re using. I would recommend taking a blank piece of scrap wood - any size larger than your puck - and drilling a hole - through the entire piece - of a tight fit diameter of a dowel that you have. This will locate the same center on both sides of the piece.
Drill a registration hole on your wasteboard of the same size. Cut the dowel length so that you can put it in a hole on your spoilboard such that it barely crests over the surface of the spoilboard, but will be able to engage your test stock.
Then use that dowel to reference your stock, cut side one, flip it, re-register center exactly on the dowel again, and then cut side two. If your dowel is tight (and your drill press is perpendicular to the stock), it should register the center at precisely (enough) the same point to prove out this exercise.
If you have the same problem, there is something other than registration / piece size wrong. But if it cuts properly, you’ve got something measured wrong somewhere or are not re-registering properly after flipping.
EDIT: Obviously, cut the side that doesn’t get cut away first, so that you have something to register to when you flip it