Depends on situation
So..
There are a few factors worth baring in mind, are you using an APC sized sensor (400/450/500/550/40/50/60/7D?) you have to remember you have a 1.6 crop factor on these cameras, so you will be getting a 112-320mm lens.
Many wildlife photographer simply do not buy a full frame camera for this reason. These type of focal lengths are alot more expensive. If you add a 2x extender you will be in the range of 640mm on a apc crop camera. Yes the image quality will not be as good as a prime 600mm lens but for anyone who is not a pro and doesnt need literaly best in the world image quality then this is the way forward. The 2x extender would in effect make a 70-200mm 2.8 a 224-640mm 5.6 lens. Unless you are an extreme pixel counter or printing above A2 the quality wont be an issue for 90% of people. I mean a 600mm f4 lens is over £6,000 and you dont get the flexibility and your wallet will have a hole the size of the grand canyon.
Also the f2.8 is a fair bit heavier than the f4, and if you do any low light photography then the f4 just doesnt cut the mustard, IS does nothing if the subject is moving, also a 2x extender will make this lens an f8. This assumption that the f4 is better quality than the 2.8 is definitely not true, the 70-200mm 2.8 is well renowned for being the best quality zoom lens in the world. Having owned both i can confirm this, the sharpness you are talking about is also a myth, the f4 with image stabilization will not be as sharp, because with IS turned on the images do get softer, its just one draw backs of IS but obviously this can be brought back with sharpening the image during processing. Also the comparison of f2.8 to f4 well yes it will be sharper comparing those f numbers, but at mid range f5.6-8 the f2.8 is alot sharper. The f2.8 uses better glass because of its aperture range hens its weight and price compared to the f4. Dont get me wrong the f4 is an incredible lens but the f2.8 gives you a lens suited to many more photographic situations. I found the f4 just not fast enough for my needs. In my opinion the f4 is more a pro-sumer lens because of its cost and f number. Not many pros would go for this option, including myself.
For me the answer is simple, it would be the f2.8 every time, but i would buy the image stabilized version, not the newest version because it is pricy (but amazing if you can afford it) it would be the mark 1 image stabilized just because at the longer focal length and the weight of the lens it does help, and you could get a very good second hand version for about £1100, if price is an issue.
For me the third party lenses dont cut the mustard. When you get into the high quality lens category like the 70-200mm the price difference becomes alot closer and make you wonder why you wouldn't buy the canon. So it would be canon lenses every time just because of there quick and reliable autofocus and that canon lenses are some of the best lenses you can buy. The extra £100 you would pay over third party lenses is worth paying. Also i assume because you are looking in the professional area of lenses you are going to give this lens some hammer and the build, water sealing and dust proofing make this lens a very good companion. I have used the third party options but found compared to the canon equivalents they just couldn't keep up in terms of usability image quality was alot closer but its how the lens performs, speed and autofocus are always downfalls of third party lenses.
Hope this helps