Not using FRAPs, no.
FRAPS records directshow, or things like games, code controlling what shows up. The text in the film viewer is just a field with text that scrolls. It's probably using an everyday windows display object.
If you can find a desktop recording software (which don't normally work with games) you could capture that as it scrolls, crop it, and overlay it into your video.
I know, that's a lot of extra work, but I *have* considered it in the past. I also miss being able to see clipboards!