Ok guys, here's a more substantial update.
I actually talked to one of the Eventide technicians last week on the phone. Unfortunately, they were not able to duplicate the kind of noise that I showed in my video. We were both pretty baffled at that, and we talked a while about potential problems, and we couldn't really come to a good conclusion. The guy told me that my pedal tested well within their "normal" range for their diagnostic tests, which included a signal-to-noise ratio test.
So, try as they might, the Eventide guys couldn't find anything wrong with my pedal, and hence did not make any changes or repairs. So it is on it's way back to me, and should be at my house tomorrow.
I have a few things I'm going to try… the first thing being a different power supply (which I was told shouldn't be the problem, but we'll see). So once I play around with it, I'll give another update.
I know that this pedal is less noisy in the loop of an amp, but I don't use pedals in a loop. So if it ends up that this is just normal noise for the pedal, then sadly I don't think I'll be keeping it.