That's not dumb, it's just the first step in you becoming an expert
You have a fixturing problem, we call it. Yes, you could use a sacrificial spacer under your part. But, it won't hold the cutout. You've got to control everything. That means, any part that gets cut loose, needs a place to go, or else, it must be clamped down so it can't move either. In many cases, it is better to reduce the cutout to chips, but in this case, you want to save it because it is your part. So, stop cutting before you get to the last line of the cut, and apply a hold down clamp of some sort. Then cut it.
Or, you could leave a tiny sliver of thickness along the bottom of the cut, joining the part to the stock. After machining, then you can cut this out with a knife or some shears and remove the remaining "tab".