So your camera has some error, everyone’s does. I would search for two major topics:
One is jigs. You basically make a reference cut first, then slot your coaster into it, and then do your engrave. Once you setup the jig, ignore the camera, because the alignment is guaranteed exact at that point.
Here’s a post I made about jigs ages ago:
The second topic is camera calibration. It’s a standard job you can run that will help make your camera more accurate.
Here’s the thread about calibration:
Be warned, calibration can make your camera much much better, but nothing will ever be as accurate as jigs. If accuracy is paramount, then reach for a jig.
Oh also, check out “set focus” by searching the forum. it can help too.