Ooh! I think I have an idea for this!
That old friend EbSynth.
First turn your clip into a png sequence.
Save a copy of one frame from the not-faded end of the clip in a separate folder. Don't edit it at all, it is your keyframe.
Set EbSynth to start on your keyframe and work through the clip. Run it.
The output won't look perfect. Import it as a clip and line it up frame perfect with the original, but on a layer above it.
Use blend mode Luminosity on the new clip from EbSynth, and it should make about half or possibly more of the fade go away.
However, it may introduce new problems. Your mileage may vary.
Edit: if pleased with results, experiment with brightening the keyframe to see if that might make the frames even closer to black into something more usable?