Ok, so I finally figured out what was causing this. In my last round, I pulled out my Destroyer to throw it for a couple holes, and the first one that I threw it on I felt the problem. It appears that occasionally when throwing the destroyer, my pinky and ring fingers will slip off the rim, leaving me throwing with my middle and index. When this happens, I can REALLY feel the torque that gets put on my middle finger, and it pretty much instantly started hurting. It took me two of those throws that day to realize that this is what was causing the issue.
Previously when I had I hadn't really thought about it, only thinking that I had got a little better snap because I could feel it more. Fortunately that round I caught what happened. Miller's recommendation was to modify my grip a bit, allowing it to sit deeper in my hand and my fingers to get a bit more traction on the rim. I'll be trying this over the next few rounds. I know it's possible, as I don't always have pain when I throw the destroyer. If I continue having the pain though, I'll probably just end up dropping the destroyer, which is sad, as it's a great disc.