Assembling acrylic pieces inside an epoxy-filled frame produces nice results:
The gotcha: epoxy oozes from between the layers to form a slobbery edge.
I tried introducing a similar coaster to Mr Disk Sander with reasonable results:
The coaster on the bottom has its original generous epoxy slobber around the acrylic disks.
Assembling the layers inside a mold seems fraught with messiness, particularly if I eventually want to get it out of the mold.
Using a finer abrasive disk would certainly help, but the whole process requires intense concentration and is utterly unforgiving of mistakes.
I figured I could attach the coaster to a lathe fixture and turn the rim, so I made a fixture from scrap acrylic:
The lathe chuck inside jaws fit inside the hole and I set up to turn the OD to a nice even diameter:
The fixture sat flush against the middle step of the jaws with plenty of clearance from the outer step, so I could turn the OD without whacking the carbide insert.
I planned to grab the OD and turn the ID to a (reasonably) concentric finish, but the outer jaws have an absolute diameter limit a few millimeters less than the 4 inch = 101.4 mm coaster OD.
After some increasingly desperate attempts, I concluded that, lacking a 4-jaw lathe chuck, there was no way to mount the coaster on the fixture and have it sit it even approximately centered on the spindle axis.
I do, however, have a 4-jaw chuck for the Sherline mill, normally used with the rotary table.
Next up: Round 2.
The Smell of Molten Projects in the Morning 
Re: Carbide insert on Sherline lathe
I saw that you’re using an insert tool. Do you use that to cut metal too? It’s appealing to me to be able to swap inserts rather than grind a HSS tool as I usually do, but I think I’ve seen it said that HSS is better for cutting aluminum (which is what I’m usually cutting).
That led me to wonder whether HSS inserts could be a thing. Probably not or they would exist?
For my simple / low stress / noncritical needs, carbide works fine for everything from plastic to steel. I have some Warner HSS inserts, but never had enough motivation to swap ’em in:
https://littlemachineshop.com/products/search.php?tabName=Products&term=hss+inserts
A shop-made quick-change toolpost came with the South Bend lathe, so IMO a QCTP / toolholder is the only way to fly! Haven’t ground a bit in a long while and don’t miss the experience hardly at all.
Ah. That’s great. Thanks.