I would certainly like the extra reach of the 70-300, but I'm afraid it won't be as versatile with the smaller max aperature. The 70-200 2.8 IS II is nice and fast, but it's expensive and won't reach quite as far.
I've got the 70-300L, and while the 70-200L is of course the "money is no issue" pro and superior lens (has an edge in sharpness, and of course constant f2.
. And if you've got the cash it's not that hard to decide for you, but my 2cents to remember anyway...
* Weight! Go to a shop, plug on a 70-200L + a 600rt-type flash and leisurely hold it to eye level for some time or dangle it from your hand. Right, that's why I got the lighter 70-300L for all day outdoor shots which creates much less torque on your wrist because the weight is nearer to the camera body.
* AF: On the 5d3/1dx the 70-300L only has horizontal af points to the side, so that's a drawback (see the manual). But af precision of the 70-300L is very good on the 5d3 anyway, even compared to the 70-200L: http://www.lensrentals.com/blog/2012/07/autofocus-reality-part-3a-canon-lenses
* Aperture: The 70-300L is a outdoor good light lens. full stop. But then it excels, because the depth of field @300mm and f5.6 is extremely thin anyway, so in may cases you wouldn't want to stop down for this reason. Shutter speed should be no issue, esp. on full frame with higher iso capability. If you need a fast aperture it is a feasibly alternative to get some also light prime lenses in addition, word is you cannot beat a 135L portrait with the 70-200L, though the latter of course is more versatile.
* Zoom lenght: Standard advice is that a tc is for occasional use and not for always on (drop in sharpness, slower af). If you don't think 200mm is enough, get a 70-300L or wait for the new 100-400L. Btw, you can plug on a Kenko 1.4x tc on the 70-300L to get to 420mm @f8, iq is good and non-mission critical af in good light is ok.