Screen capture (H.264, nonstandard resolution) is blurry in FCP: I need to fix this, but how?

[Jeff Meyer]

I would first suggest hitting the H.264 with ProRes422. Use Compressor or Streamclip.
Then I would go into Cinema Tools and conform the 30fps 29.97 so Final Cut isn’t doing rate conversions. This simply changes the playback rate, no conversions, no rendering. Cinema Tools won’t work with H.264.
Lastly, on your Sequence Settings, Video Processing tab, are you using BEST on the Motion Filtering Quality?