AMD's uninstaller wasn't cleaning the driver out well enough. I mean both the one in the CCC installer and the AMD Uninstall Tool. They were leaving dregs behind that were keeping it from installing properly.
Using Guru's driver uninstaller and reinstalling fixed the problem. It cleaned everything out, and I was able to access the settings after a reinstall.
So basically, AMD needs to update their uninstall tools so that they actually work properly. They don't clean up the remains of the uninstall well enough and it was the cause of this problem.