Lacking a 4-jaw chuck for the lathe, this should suffice:

Which is just the Sherline 4-jaw chuck chucked in the lathe’s 3-jaw chuck, with both chuck Jaw 1 positions lined up and marked on the acrylic disk fixture. The picture is a recreation set up after the fact, because I lack a good picture of the overall scene.
Now it’s easy enough to center the fixture, stick the coaster in place with reasonable accuracy, then tweak the Sherline chuck to center the coaster:

Because the bottom layer is a laser-cut disk, eyeballometrically aligning its edge to a simple pointer worked surprisingly well:

Turning the OD down to match the bottom disk meant I could finally get decent results with zero drama:

From the bottom, this one has a 3 mm mirror, the 3 mm fluorescent green frame + petals, and a 1.6 mm top sheet:

This one has a 3M double-sided tape with low-surface-energy adhesive layers between the mirror and the fluorescent blue frame + petal, with epoxy between the top layer and the frame.

If I never tell anybody, they’ll think the slightly granular look if the tape was deliberate; it looks OK to me.
And, for completeness, the crash test dummy from the start of this adventure:

I don’t know how to avoid the bubbles, as the usual torch-the-top and pull-a-vacuum techniques pop bubbles at the epoxy-air interface. These bubbles are trapped under the top acrylic sheet, even though I was rather painstaking about easing the layer down from one side to the other while chasing bubbles along.
Maybe I can define bubbles as Part of the Art?
Definitely fancier than chipboard, although not nearly as absorbent.
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