I have been trying to crack this puzzle for some time and I think I have figured out a pretty good solution! I love that I can record audio and webcam video straight in the SpeedGrader, but I have needed to have the ability to record my desktop instead of my screen. Well countless of hours and too many downloaded apps led me to CamMask (http://www.cammask.com/), a free virtual webcam software. Now this program does way more than I need it to, but the one thing it does is trick your computer to thinking a feed of your desktop is actually a webcam. This allows us to get the option of a desktop recording in the SpeedGrader under the same drop down as your webcam! The other nice thing about the CamMask software is the small watermark on the free version and the low cost of the paid version.
Here is a video I put together of me setting this up and explaining it in more detail (sorry about the static).
One note on using this method, it takes a bit for the videos to process. This is problematic because there is no indication through Canvas on how long it is taking to process, or if it is even still processing. Take a minute to jump over to the "HTML Video in Canvas" idea that is open for voting right now Ståle Heggset. Up-vote this idea so this feature will become more user friendly.