I would always say that you would be better off using a higher ISO in the camera, so that the resulting image was a little bright and you then had to reduce it a little in post, than to be a bit under exposed because of too low an ISO.
I'm not sure where the 5DIII sensor comes, it seems some of the latest sensors are not so bad with this, but I would also only ever shoot with the camera set to full stop ISO values. In general Canon cameras use analogue amplification steps to set the full stop ISO values, which is what you want. The 1/3 stop, and the expanded ISO values are created by digital manipulation after the values have been digitised.
For the 1/3 value above a full stop value it literally digitally boosts the exposure by 1/3rd stop, just about the worst thing you can do for noise. ISO 400 is in this category. The 1/3rd below is pulled back, which is kind of what you want, but unless you absolutely need to have JPEG files from the camera, such as for sending off the a publisher at half time in a match, I would be shooting in RAW anyway, to offer optimal post processing.
So I would most definitely be shooting at ISO 6400 not 4000, but with the same 1/250s ƒ/5.6 to ensure that I got an optimum exposure. At least it seems that you are not running into issues from gas discharge lighting, which also can screw with your exposure and apparent WB.
The real problem is that all of these indoor venues although seeming very bright to us, are actually as far as the camera is concerned very dim indeed. Coming from the days of ISO 400 HP5 and TriX black and white films, which were already pretty grainy, before you pushed them a stop, what we get now is actually pretty amazing.