1.20.1, Fabric.

I actually downloaded this mod primarily because I created a skin with a mask that I want to have to look through at the world, but the second layer does not render. The bangs being visible in one of the showcase images on Modrinth led me to believe that the second layer - such as glasses and hair - would render. Am I perhaps simply doing something wrong or should I look at compatibility issues?

That's my problem, I didn't state it in the mod description
that the gif was recorded in an old verison (probably 0.5.0) and you have to change the config to make your second layer visible like in the gif:
Here X offset is -0.06, Y offset -0.08 and disabling depth 0.08 (fov 76)

And in the next release, skin_head_3 will be added to the initial config