Thanks for the reply's. This is helping. While I am now leaning towards the 70-300, I am also wondering if my 2x tele extender (v2) will auto focus with the 70-200mm f/4-5.6L IS on my 6D? I know it doesn't on my current 70-200 non IS. Anyone know?
From The Digital Picture review
The Canon 70-300 L is officially NOT compatible with Canon extenders. This is the only Canon L zoom lens with a widest focal length of 70mm or longer that is not compatible with Canon extenders (as of review time). I've tried mounting Canon extenders to the 70-300 L - and they do not fit. The rear 70-300 L lens element physically hits the front element of the extender.
Then, in a major firmware update announcement for the Canon EOS 1D X, Canon included the 70-300 L and Canon EF 1.4x III Extender combo on a compatibility list.
Intrigued, I did more experimenting. What I learned is that the 70-300 L's rear element retracts into the lens far enough that, at about 250mm, there is enough clearance for Canon extenders to mount.
The available with-1.4x focal length range is about 350-420mm and 500-600mm with the 2x installed. Zooming out wider than the 250mm-or-so zoom ring mark results in a physical bump inside the lens. I'm guessing that it is the rubber around the edge of the extender element contacts the rear 70-300 L lens element or its barrel. I do not recommend mounting this combination due to potential damage the to lens.
I of course felt the need to try out these unsupported combinations. My solution to the damage risk issue was to Gaffer Tape the lens zoom ring to lock it at the 300mm mark while using extenders.
Mounting a 1.4x or 2x extender behind any lens reduces its aperture range by 1 or 2 stops respectively. At review time, only Canon 1-Series bodies can autofocus when using the very-dark f/8 max aperture the 70-300 L lens and 1.4x extender combination yields. Tested 1D X and 1Ds III bodies autofocused this combination.
No bodies can AF with the even-darker f/11 max aperture the 70-300 L and 2x combo yields. While it tries, the tested 1Ds III body cannot lock focus with this combination.
The interesting part of this story starts with the fact that the Canon Extenders do not report their presence when mounted behind the 70-300 L. The reported max aperture incorrectly remains f/5.6.