As if OBS wasn't good enough already - I just found out that using it, you can stream Quake Live in fullscreen. No third-party hacks needed, no FPS drops, it just works.
If you've used Xsplit before you'll probably find the UI easy to learn. If not, I recommend checking out the official quick start guide, the more in-depth official overview guide or my own quick start guide.
Once you have a grasp of how OBS's UI works, it's very easy to set up for Quake Live: http://i.imgur.com/1K2LW.jpg (1.08MB)
If you don't have 2 monitors, just switch to Windowed mode while setting up OBS, then switch back to fullscreen. Should work fine.
Here's a short example with my streaming settings (720p, 2000kbps), please forgive the awfulness of the play: http://lutro.priv.no/qlobs_short.mp4