Thanks for your reply. In the end I was able to isolate the graphic when nothing else was "behind" it and make a still out of it. I cropped the still to just the size of the logo. I then placed that in a video track above the original footage and set the composite mode for the cropped still to "difference". I had to constantly keyframe the opacity of that clip depending on the brightness of whatever was "behind" the logo at any given time. It didn't make the logo completely disappear, but it did make it look as though there was a black square in that place with about 50% opacity. In other words, you could still see the movement "behind" it. There were times, especially when the background was pretty bright, that you could still make out a slight ghost image of the logo, so I ended up putting some text in that spot, as well.
Thanks for the warning about copyright. I'm just a peon doing this fourth hand and assume the original client has permission to do this. I'm in no position to be giving them legal advice and am only doing work for hire. The legal ramifications for this would fall on them.
Thanks!