It works now! Here's what I did to fix this problem, for anyone having similar issues:
1) Go into the control panel to uninstall Catalyst and all other ATI software.
2) Go into the Device Manager to uninstall any "Display Adaptor" drivers.
3) Run the driver installation program, but cancel out of it as soon as the installation files have been extracted to the hard drive.
4) Restart into Safe Mode
5) Run Driver Cleaner Pro
6) Run Driver Sweeper (maybe unnecessary, but better safe than sorry)
7) Run CCleaner: both the hard drive and the registry scan options (again, better safe than sorry).
8) Go into the Device Manager again, right click on the standard driver that Windows installs under the "Display Adaptors" category and click "update driver."
9) In the wizard that shows up, click the bottom button, labeled "Browse my computer for driver software" on the relevant page.
10) Point it to the folder into which the driver software was extracted in step 3.
11) It should install without using InstallShield, and work this time (at least, it did for me).
A reference to the essentials of the above process can be found here:
Answer
It looks like this is more a problem with InstallShield (or rather, the way that it was implemented for these drivers, since it works fine for other stuff on my computer), rather than the drivers themselves. I hope this helps if you're having similar issues.