In your case you are adding artificial lighting into the mix and that seems to be where the vast majority of the flickering is coming from.

Some light sources powered by alternating current can vary by more than a stop between the peak and the trough of their AC cycle. And since metering is done at a different instant than exposure, a conventional meter can't accurately predict the state of the lights in their cycle when the shutter is opened.

The conventional solution is to set exposure manually and use a shutter speed that is long enough to include an entire cycle of the peak and trough that the lights go through. For lights running on two phase 60Hz alternating current, a shutter speed of 1/120 second is usually sufficient. For lights running on 50Hz, 1/100 second will usually do.

If you have access to the fairly new Canon EOS 7D Mark II, there is a feature introduced on that model that may help if you absolutely must use faster shutter speeds. The Anti Flicker mode senses the frequency and timing of the lights as they flicker and times the shutter to only release when the lights are at their peak.

I haven't tried the Anti-Flicker feature in terms of time lapse video, but I have found it extremely useful in shooting night sports in outdoor settings under the flickering lighting typically used at high school and community sports facilities. Not only does it give more consistent exposure and color temperature from frame to frame, but it also allows me to shoot at 1/3 to 1/2 stop faster shutter speeds because the lights are always at their peak when the shutter opens! Anyone who has ever tried to shoot at 1/500 to 1/1000 second under 60Hz lights know what I am talking about: half the frame is brighter and fuller spectrum and the other half is dim and brown as the lights were falling off (or ramping up) during the time the slit between the shutter curtains was transiting the sensor.

Another preventative measure would be to use a white or black background that could be blown out or crushed to give a constant level to much of the frame.

Update: For existing files that demonstrate this issue, you would need an application that allows very precise fine-tuning of exposure/brightness/saturation/etc levels. 0.05EV is 1/20 stop. Perhaps software created to grade video frame-by-frame would be the best option.