the MkII is the one you want. Shutter speed is not an attribute of the lens but of the camera and amount of light available.
f/2.8 is the most you're going to get out of a lens as far as ability to let in light, so you're at the top end of telephoto zooms with the 70-200 f/2.8 MkII. If you want longer reach there are things like the Sigma 120-300 f/2.8; but I'd stick with the Canon MkII.
Of course, technique is going to get you a lot farther than a new, $2000 lens. If you're shooting in broad daylight and getting blurry images the very last thing that's holding you back is your hardware. What sort of settings you are using, how you're supporting the camera, how you're standing, etc all have way more of an effect than a new lens.