I do not have a III or Aperture, but have you tried any other software? Possibly DPP that came with the camera or you could try a trial for LR or PS and see if it is doing the same thing elsewhere.
One thought is that in the in-camera settings (picture style) have the brightness too high. This would result in the jpeg preview appearing fine on the back of your camera, as well as the pre-processed preview you see in LR.
Do you have display brightness up high on the 5D3? This could lead to a false exposure assessment.
This would not result in the software showing a brighter image before it shows the actual image. It would be a picture style setting. But I do fully agree about using histograms. Though aren't histograms based of the jpeg preview as well?
One method I sometimes use to get a better idea of seeing how the RAW may look on the back of the camera is to make a custom Picture style with contrast and sharpness all the way down and leaving everything else at default. I don't do this much anymore as I now either have it set to monochrome as I do all my personal photos in B&W and like to see it in B&W on the back of the camera (shoot RAW so color info is still there) or when I shoot sports, I have it set to Landscape to get an idea of what my processed version might look like.