Remember, if you opted for a 1.4x teleconverter, you'd probably lose less sharpness with the 5D - but the 7DII will probably still pull in a hair more detail than that setup.
Perhaps the boost in high ISO performance on the 7DII is more than adequate for you - it sounds like it may be. You could always go around the internet and find other people's shots with both setups for more direct results... I think you may be better off with the 7DII if you're focal length limited and can live with the high ISO performance of the 7DII, then you should be happy with that combo.
So, the real question would be if you'd need more lens to run the 5DIII, and it sounds like you might.
If you can't add a 1.4x teleconverter or get another lens, the 7DII is probably the right choice for you.
If you can add a 1.4x teleconverter, that would give you ~280mm vs ~320mm EFL with 10% more resolution and still a ~.5 stop ISO advantage, but slightly worse AF performance from the teleconverter. That's a wash, I think.
If you are NOT focal length limited and can frame shots as you please, then the 5DIII will be at an advantage. But I doubt this is the case with 200mm unless you're only shooting indoor sports.
If you're bouncing up against ISO 6400 on the 7DII a lot, you might end up wanting the 5DIII... or a faster prime lens if you're shooting indoors and don't need 200mm.
Seems like the 7DII is generally going to be the superior choice for you as long as you are somewhat focal length limited, but if you're NOT FLL, you should strongly consider the 5DIII.