First things first, the Final Pass Exponent value simply decreases/increases the actual value of the Final Pass Density multiplier by an order of magnitude. So if the Final Pass Density is 5.0 and the Exponent is -1, it results in a value of 0.5 used to multiply the Density coming from the particles. Thus, changing it to -2 produces 0.05, -3 is 0.005, -4 is 0.0005 and so on. Of course, the resulting Alpha of the render image will be 10x lower when you render with -4 as opposed to -3, so no wonder things look faint and faded.
You should NOT manipulate the levels of your input in AE. Just remember that the render output from Krakatoa is PRE-MULTIPLIED. So when you render in Krakatoa, leave the background black, and when AE asks you on import, answer that the image is Pre-Multiplied and keep the default black swatch value - it will divide the RGB channels based on the Alpha and Background swatch values and remove the black from the RGB components.
From Krakatoa, render with Occluded particles pass so you would have a background layer containing the particles behind Matte objects. Put the background image at the bottom of the comp, the Occluded layer on top, the Geometry rendering from the non-Krakatoa renderer above it, and finish with the main Krakatoa render output which contains the foreground particles.
Keep in mind that bright particles on black background will generally appear brighter than when composited over a bright background. For testing purposes, I would suggest setting the background in Krakatoa to a brighter shade of sky blue to get an idea how little they actually contribute to the color. When rendering out the final pass, disable the background override and save with black background to make the pre-multiplied post-division easier in AE.
Here is an occluding teapot in a spherical volumetric particles cloud with Final Pass Density of 1.0 Exponent -2:
Here is the same thing with sky-blue (0.2,0.5,0.8) background:
This is the geometry rendering of the teapot itself:
The actual Alpha in the saved image might appear relatively high when looking at the channel in the frame buffer. In my test, it was around 0.5, which means that 50% of the background image would be blended with 50% of the Krakatoa particles’ color. However if both colors are rather bright (in my case a mid-gray over mostly a blue sky), the particles won’t look too solid - they will be literally 50% transparent.
This the the Alpha:
When composited in AE over a photo of the Olympic Fire in Vancouver I took a few years ago, it produces this (ignore the wrong pespective ):