I was able to identify the toilet that we have as the Thetford Aqau-Magic Style II, albeit mostly by matching up pictures on the Thetford website. Switching to Amazon, I found a replacement ball and seal kit that I could get next day on Prime (first time I've been able to use the new free next day shipping service).
Thetford 34117 Waste Ball Kit for Style II / Style Lite / Style Plus
Unfortunately, after receiving the kit, which Amazon was kind enough to drop on my front porch, I found that the instructions only cover the Lite and Plus models, not the II.
Fortunately, the lack of instructions is not usually a deterrent for me, so I proceeded with an ultimately successful installation attempt, which I conveniently captured most of on camera.