I'm a bit puzzled as to why your 70-200mmL f/4 IS has issues at 200mm. It is much sharper than the 400mm. Get it fixed, its one of the best lenses for the price.
It s pretty well known the 70-200 f4 IS perfoms poorly when at 200mm and at or close to MFD. I have it and this problem is noticable. Mine also suffers from the dredded slipping focus problem. Buyers beware to check for this issue if buying new or second hand. Sorry for going off topic. Otherwise its a very sharp lens.
I'm with Mt Spokane on this one, and "well-known" to whom? I owned the 70-200 f4 IS and it was as sharp at 200mm as any other focal length and even with the 1.4x at 280mm, it's sharper than most other lenses. I think you definitely need to have your lens calibrated/repaired. Here are some results at 200mm from DxO, LensTip, and Photozone, all showing the lens to be extremely sharp. Now back to the 400mm 5.6...
The weakness of this very good lens as I stated earlier is at MFD. "Well known to whom"? Google is your friend.