Why did you not look for the piece in the pattern when you took it off?? It was probably stuck to it being that is how it came apart. Did you use blue tape under the pattern or some other backer?? That is why it is a good idea to not adhere the pattern direct to wood. I use to to do that when I first started but finding out about blue painters tape on some forum many years ago I have not looked back. I have had small fragile pieces break but almost always just glue them back on. If the original piece it is hard to tell the difference. Good luck
By the way with that patch trick, the easy thing to do is use a sharp exacto knife and score the piece instead of using a pencil and cutting. The thin blade is as sharp of a point you can get. You would not even have to use the saw to cut it. A few passes over it and pop it off the plywood.