I can tell you how i did it, but i ended up buying a capture card, and finally after that i still wasn't satisfied so i ended up buying two Nvidia GTX 780's which i run in SLI, and with SLI you can actually clone your screen while having 144hz on monitor one, and 60hz on monitor two.
What you want to do with AMD, is just setup crossfire like how u did. Then you want to open up dxtory, which is a neat little program that let's you capture gameplay. But we are not going to capture gameplay with it, we only want to use the dxtory directshow output. I could write a tutorial on this if you want.
Now when u have that setup alltogether, you want to open up VLC mediaplayer and open that directshow output. You can then drag this to a second monitor and enter fullscreen mode. Then you can use that second monitor with the images to stream.
This is a very dirty workaround, which you will need a very strong CPU for. I then hooked up this second screen with the fullscreen VLC up to a capturecard to stream this on a different computer, because i had issues with this.
As for now, AMD simply doesnt support going in CFX-mode with the screenclones. With the above option it is possible to capture the gameplay at lets say 30 fps, and then output it on a second screen with VLC so you can use that for streaming. But still, i experienced a drop in frames because that directshow thing actually takes up a bit of GPU/CPU power. I found the best results were to limit the output to 720p 30 fps. When you go to like 1080p it takes up to much FPS.
This is the exact reason why i ended up buying two Nvidia cards because i want to be able to play on 144hz, without any framedrops when duplicating the screen. And now it finally works :D
Oh and indeed: Crossfire disabling isn't OBS's fault indeed. It is because AMD's drivers don't support a screenclone while in CFX mode. Crossfire want's to have one screen in fullscreen in order for it to work. That's the main reason why Crossfire is disabled when you are going to switch to windowed mode in game. Don't ever do that when you have crossfire enabled because it will actaully disable it.