You have a 200-400 f/4 corrected for f/4 @ 200mm which is £9k, to correct it for f/2 @ 200mm is going to be expensive and heavy. These superteles were made to be used wide open and if it was plausible to use them at variable apertures someone would have made one. The Canon 70-300 f/4-5.6L IS looks to be your best bet if you want a more professional variable aperture zoom and even that was about £1500 at launch. I use the 200 f/2 with TCs which can take it to a 280 f/2.8, 340 f3.4 or 400 f/4 so the versatility is there while keeping the weight, complexity and price down at an acceptable level.
No, the 200-400mm f/4.0 already has most of the necessary optics to correct aberrations at 200mm f/2.0 because it is more or less the same thing. So there is no more weight or expense to add.
You have to realize that variable aperture is the natural unmodified functionality of a zoom lens in its most basic form, a fixed aperture zoom has been purposely and artificially limited by a mechanical coupling to prevent the lens opening up to its widest aperture at the shorter focal lengths. There is nothing special about a fixed aperture zoom, it has just been contained not to fully utilize the light gathering potential of the front element when not used at the far tele end.
Secondly telephoto lens complexity is related to the physical aperture size and the diameter of the front element, the actual f-number is irrelevant.
Instead of thinking about correcting a 400mm lens at f/4.0 and a 200mm lens at f/2.0, you need to be thinking about correcting a lens with a 100mm aperture.
The fact that the 200mm /f2.0 lens behaves so well with TCs up to 400mm f/4.0 is just proof that that a 200-400mm f/2.0-f/4.0 is perfectly feasible with little added cost or complexity.
The new Nikon 80-400mm AF-S is another great example of allowing a zoom lens to utilize the aperture of the front element at wider focal lengths. It is more or less as sharp as the 200-400 but a stop slower at 400mm, less so at the wide end.
My eyes are on Sigma making a variable aperture 200-400mm f/2.8-f/4.0 or perhaps a 200-500mm /f3.5-f/5.6.
Both Nikon and Sigma have released F/2.8-f/4.0 variable aperture zooms at shorter focal length requiring retro-focus designs. A variable aperture telephoto is much simpler than that!