I've not shot with a 5D3, but I can vouch for the 70-200 II + 2xIII on a 1DX. I tried it at a youth-league football game in evening, rainy, overcast light. Uniforms were grey/black. AI servo, AF mode 4 (erratic movement) untweaked, 'center 9' AF points selected. The kids weren't nearly as fast your cars, but they generally follow less-predictable paths. OOF rate <1% until they pass MFD.
With the 2xIII mounted, the initial focus lock feels substantially slower, but the following tracking felt (and worked) just fine. I'm not in the motorsports scene, but 2xIIIs are cheap to rent - definitely encourage you to try one in your working environment.