So I'm not even sure how to relay this, but here's how it went:
I logged into KoL via FF, and ran the test. Sure enough, it happened. So Veracity was correct. However after the first time, it behaved normally. Even after logging, closing the browser, relaunching the browser, and trying it again the issue didn't occur.
But when I logged back into Mafia, fired up the relay browser and tried it, I got the issue with the overlay. So I logged out of Mafia, launched FF again, and repeated the test. It behaved normally. Log back into Mafia, and the overlay problem occurs. Weiiiiird.
In any event, I've logged the bug in KoL since it did happen there the first time I tried it. Thanks for all the help, and Veracity? I guess I owe you unnamed stakes!