NEVER MIND! After hours of troubleshooting, I discovered a post buried on this site that suggested installing an old theme (Twenty Eleven).
So I deactivated all of my plug-ins first (just in case) and then installed & actived Twenty Eleven. Clicked the Customize link, and it worked. I’ve no freaking idea why it worked, but it did. I’ve now reactivated my preferred theme, and it’s still working…
Will slowly reinstate my plug-ins, but at least for now a crisis has been averted.
Thanks to the person who posted that remark about Twenty Eleven!