My thanks to Chris Pratt for the near-perfect Handbrake preset.

One of the developers of HandBrake suggested that the AppleTV preset in HandBrake with the quality slider bumped up to 62% gives a near perfect encode. What the thread didn’t mention is that you’ll need to uncheck “Large file size”, as well, since this is only for encodings where the produced mp4 will be greater than 4GB and breaks compatibility with most software and devices when enabled.