I think I might have it.
Sikuli, an MIT project, uses the CV library to pattern-recognize places to click. (I'll add links later, or you can Google it.) LINK
It's a bit flaky on Ubuntu, but I got it working out-of-the-box on XP. Which makes me sad, but also gives me progress.
Well, Sikuli on XP on VirtualBox on Ubuntu on old Dell hardware. This is not a fast process. But it seems to be working.
Not a replacement for having access to the API in the first place, but as it doesn't require me to a) decompile a program and b) code Java, I'll accept it.