Trying a sequence of offset and raster filling produced a credible 0.5 mm deep inlay pocket in a chunk of acrylic:

The odd pattern inside the pocket comes from the offset fill:

Combining the two fill patterns produces a smoother bottom in the pocket, but it’s a good thing nobody will ever see it from the back side, because the offset fill concentrations chew through the entire 3mm sheet.
A spade-shaped adhesive sheet bonded the veneer into the pocket, after which I sanded the surface flat with 200 grit sandpaper and hit it with some polyurethane sealer. Next time I’ll knock the finished surface flat with 400 grit paper.
The gray acrylic disk has a rebate around its rear rim for the “gold” ring, with the adhesive disk gluing the black disk to the front:

Which looks like this when it’s stuck together:

I don’t like the raggedy surface of the rebate above the ring, although it’s not too awful in person.
Having gotten reasonable results with the acrylic pocket, I cut an acrylic spade and stuck it in place with another bit of adhesive sheet, leaving it standing proud of the surface:

The red earring used up one of the hundred slugs left over from the capsule plate; I’m ready for a surge in demand.
The adhesive sheet is workable in disk form, but the spade shape is entirely too fiddly. A dab of acrylic adhesive would likely work just as well and be easier all around.
Stipulated: they’re cute, but not that cute.