All the big whites in vII are excellent lenses. If you haven´t actually used them, I would recommend renting your most likely candidates first. It can be expensive gambles.
I have only tried the 300 f2.8L IS II one weekend and I believe it is the fastest AF of all the big whites. IQ is excellent and, as far as I can tell, it is the one who gives best IQ with the 1.4xIII and 2xIII extenders. It is the least expensive (the word cheapest sounded totally misplaced here) and the smallest in size and weight, giving you 300/2.8, 420/4 and 600/5.6. Downside is if you need more reach.
The 400 f2.8L IS II gives you excellent performance on its own, very good with the 1.4xIII, but is in my view a bit soft with the 2xIII. The results are still very usable though. 400/2.8, 560/4 and 800/5.6 is an impressive versatility. I had this lens, but sold it. I kept the 600 f4L IS II and bought the 200-400 4L IS 1.4x. But I question whether I should have kept it or not. This lens is significantly lighter than the version I, which made it an alternative to me because of an option to handhold. But it is significantly bigger and heavier than the 300.
I had the version 1 of 500 f4L IS. Its main benefit is being smaller and lighter than the 400. But with f4 it is 100mm shorter than the 600 and one stop slower than the 400. So I decided against the version II of this lens when I updated.
The 600 f4L IS II is an awesome lens, which I believe I´ll keep until I´m unable to carry it. Performance with the 1.4xIII at 860mm/5.6 is excellent and actually better than the 800mm f5.6L IS. At 1200mm/8 the problem is first the very narrow angle of view (you need to find whatever you want to shoot in the viewfinder) and thereafter AF. With the 5DIII and 1DX you only have the center AF point available and in my experience, that requires very stationary objects. It is the biggest of them all, about as heavy as the 400/2.8 II, it requires a big backpack and it is almost 2x the price for the 300/2.8II. When I sold the 400/2.8, I kept this one.
The new 200-400 f4L IS 1.4x zoom is not a prime, but it is phenomenal. It is very versatile, with its 200/280 - 400/560 zoom range. about as heavy as the 400/2.8, but smaller. IQ is slightly behind the primes, but the difference is academic. The zoom function gives you a little less need to crop, to make up for it. AF is a fraction slower, but still very fast.
Summing up, these lenses are the best in the business for their respective focal lengths. They are big and expensive though. Make sure you know what focal length you need before you buy. But I can promise that when you unwrap your very own big white for the first time, it gives you that "Yes!" feeling