Let's say my settings are ISO 3200, 1/125 at f/1.4. Those remain absolutely consistent, because I've set my exposure and don't need the meter anymore. No matter what the meter reads, my settings are ISO 3200, 1/125 at f/1.4. I'm in manual mode. The meter should have absolutely no bearing on the speed of operation of the camera. That it does is absurd.
...and that's the issue, right there. Even in M mode, the meter provides a reading, as you state above. In the 7D manual, Canon acknowledges that the frame rate may drop in low light (bottom of p.93), and although they don't explicitly state it, that's due to the metering, which needs to integrate the light input over a longer time for a more accurate reading in low light levels. As I stated above, I'd expect the same issue with the 5DIII. Note that Canon appears to have taken a different approach with the 1D X - instead of integrating the same number of zones for a longer time, in low light the 1D X metering switches from it's ful 252 zones to only 35 zones, integrating the input spatially instead of temporally, which I suspect is intended to allow the high frame rate to be maintained even with metering before each frame.
The bottom line is, as the software guys like to say: "It's not a bug, it's a feature." Most people shooting in M mode would like the camera to meter. Most people would like the camera to meter even in low light levels. So, it meters before every shot, and that takes fractionally longer in low light. The camera doesn't know if you're going to shoot one shot, or mash down the shutter button for a long burst. It just knows it's supposed to meter before the shot, and not release the shutter until the metering is done...unless you tell it otherwise.
As suggested above, using continuous AE Lock (whether holding the AE Lock button during the burst, or reassigning the shutter half-press to AE Lock) will stop the camera from metering between frames of the burst, giving you the full frame rate.