If you're actually using your choice of lens for any significant time interval, you'll hate the 70-200mm and will prefer the lighter option, unless you have the privilege of standing there with a monopod like some of the fat photographers do.
Another consideration is you simply can't move fast enough with the 70-200mm (wait before responding, "I can!"--what I am saying is that no matter how fast you can move with the 70-200mm, you can move much more quickly with the 200mm f/2.8L prime lens).
I would strongly urge you to buy another body and put the 85mm f/1.8 lens on it, and use that with your existing camera and the 200mm f/2.8 L II. If you're really going to be working a lot with the 70-200mm, the two-camera alternative is way better, and both lenses are sharper at comparable apertures than the 70-200mm.
That combo produces much more success for me over the past million photos or so than the 70-200mm ever did.
I had the 70-200mm for a while and hated it all the time, and sold it for $800. I have the 70-200mm f/2.8L IS version II, but it definitely doesn't get nearly as much use as my 200mm and 85mm prime lenses.
Some things the 70-200 just isn't even an option for. I dread even the thought of trying to cover fast-moving two-hour sporting events with it.
Ok, the 70-200mm is "effortless" to hold, don't get me wrong... but over two hours I would take the 85mm, 135mm, or 200mm primes every time, every time.
And even if you have the zoom lens, getting another camera to put something else on so you don't have to switch lenses is going to save you a lot of fooling around on site when photographing something that's really happening. If you shoot only contrived and posed subjects, I guess this doesn't apply. But normally nature, animals, and people definitely don't wait.