Well, I managed to fix it. I assume it's some issue with the procedurals or something. I tried applying a different material to the eye surface; that didn't work.

So I applied the eye material from another character, then went into the Material Room and swapped out the texture map only. It rendered fine that way. Eye color I wanted, no weird white rings. So I assume the issue is somewhere in the spaghetti of material nodes.

Love the results with SuperFly, but it's got quite a learning curve. Spent half the day trying to figure out why I was getting weird blocky shadows in one render. Turns out, I needed a higher "Max Transparent Bounces." I had it at 2, and 8 or 16 works a lot better. I have no idea what "Max Transparent Bounces" even is, but too low a number has ugly results in certain situations.