I got some help on the 3DCoat discord group, and it sounds like the functionality that I wanted does not exist, at least at the time of this writing. I'm following up here in case someone else is searching for the same thing in the future. I recorded some gifs showing the Substance Painter's functionality that I wanted to replicate in 3DCoat, and3D Coat's behavior in the same circumstance.
Substance Painter: Each object has its own material, texture, and UV set. Each texture has have several paint layers. The paint layers of each texture/material/object are completely independent of the others:
3DCoat: I have multiple UV sets but each UV set uses the same texture and layers as the other UV sets. This isn't work-blocking, but it means that with 3 different UV sets I have 3x the amount of layers in one file, and it gets confusing to keep track of. It would be nice if I could automatically hide the layers that were not being used for whatever UV set I was painting on. One good thing about the way 3DCoat does it is that you can seamlessly paint from one UV set to another, whereas in Substance Painter since each UV set gets its own dedicated material and texture layer stack, you can't cross a paint stroke from one UV set to another.
