Interesting setup, I'm sure it would work...again the main concern, is that you test and set the AFMA at the average distance you're shooting targets (assuming any AFMA is required after you get the lens back from Canon). If that average distance is over 100 feet, you might need a setup with slightly deeper-spaced targets to measure the focus accuracy.
Regarding the tree limb image...I assume it's heavily cropped? If you shot it at f/2.8 (or with the 2x TC, you're saying it's a 600mm image at f/5.6??), then it's possible that is getting near the full sharpness of the lens. It's hard to tell. It does look like some of the limbs are softer than others, by a bit...which is normal (and even desired) at a wider aperture.
One thing is certain...the 2x T/C, even the new series 3 one...will indeed produce an easily noticeable softness at the pixel level, when compared to the 300mm lens with no T/C. This is clearly shown in canon's own published mtf chart comparison. Also, certainly autofocus speed and accuracy will be affected, especially in servo mode. The T/C that is supposed to be the better compromise, is the 1.4x iii. But of course that is "only" 420mm, instead of 600mm.
Again, (I digress but here goes...) The 500 f/4 that I rented (version 1) back in 2011, I used combined with a rented "new" 1.4x iii T/C. The results at 700mm (mounted on either carbon fiber tripod or monopod, and via manual focus live view...since AF would not work accurately no matter what)...were ok for maybe an 8x10 print (when closed to f/8), but that's not saying much. Certainly it was softer than what you have posted so far. Of course this was on a 1.6x "crop" 50D...so full frame "equivalent" was 1120mm. However, as I have realized after buying the 6D, it's hard to equate the difference from one to the other, since the sensor and processing of the 6D, are so vastly superior...that it really does seem it has more than 20 MP resolution...especially in the lower ISO range. Even at say ISO 2500, it doesn't lose as much resolution as the 50D loses, at ISO 800. I might even venture to say the 6D's sensor is superior to the 5D3's. Of course the 5D3 starts out with a few more pixels. I assume their processing is "identical"...but surely there is some tailoring. Of course the 5D3's AF in good light is far superior (and apparently to most every other camera in existence save for the 1DX, and including the 7D), but that is pretty much common knowledge. It's in very low light, where the 6D's AF is either superior, or at least doesn't give up anything to the 5D3. Which is good, since the sensor's output at higher ISO is noticeably better, in my experience.