AME (and Premiere Pro) does not support color management in the way that After Effects does. Via Dynamic Link, which is how AME reads After Effects comps, the color-space-adjusted pixels are not corrected for screen display.
To get the results you want, add an adjustment layer to the top of the layer stack in the comp and apply the Color Profile Converter effect. Set the Output Profile to Rec.709 (sRGB is practically identical and will also work, but Dynamic Link uses Rec.709 internally so is a better match). This forces After Effects to transform the adjusted pixels into a non-linearized color space that looks correct.
Note that while the CPC effect is active and View > Display Color Management is enabled (it is enabled by default), this extra layer of color transforms will make the comp appear incorrect in After Effects, at the same time the comp will now look correct in AME or Premiere Pro. Disable Display Color Management to make the appearance of the comp in After Effects match what you see in AME or Premiere Pro. While working on the comp, however, you probably want to work with Display Color Management enabled and the adjustment layer disabled.
Under the hood, when color management is enabled in After Effects, the pixels it writes into the cache include the appropriate color transforms for the settings you have chosen. When the comp is displayed in the Composition panel in After Effects, an additional transform is added to the screen buffer pixels (not the pixels in the cache) to make it look correct on your computer screen, or not if you have disabled Display Color Management. When the pixels are read through Dynamic Link, no display color management happens, nor does AME or Premiere Pro apply any, so you get the same appearance as having Display Color Management disabled in After Effects.
Make sense?