@jaz I have conducted your requested experiment. I adjusted the design slightly to give you as much information as I could think of. What I observed is that the circle would not turn until I broke it free of the mating part. It did not cut completely through.
Then I turned the circle and while it turned inside its hole, it did not stay centered. You can see the change in the photos below. Each picture is at about 45 degrees from the last.
The next test was to see if I could isolate the error to a single axis. The slots in the base are cut to be a tight fit, with very little wobble. I estimate the wobble at about 0.2 degrees.
I see a variance of about 0.5 degrees in the X axis alignment, from perfectly square to 0.5 degrees. This gives an approximate X axis error of 0.25 degrees. The tilt is top to the right of the laser.
At the base of the insert I measured a gap of 0.075 inches. The tab is 2.09 inches tall at its inside face, measured vertically. A little trigonometry gives an error of 2.055 degrees. With the estimated wobble, I will just call it 2.0 degrees.
I did not see anything on the head or the head support that would lead to the top of the head being tilted toward the back of the laser by 2 degrees.
Please let me know if there is any additional information you require.
Thank you for your time and attention.
:edit: Adjusted error value on X-Axis