MPEG2 Program, Transport, Elementary Streams?
Elementary wouldn’t have audio.
I’m not sure of AME’s ability to handle all the above nor what version you’re using.
You may want to test with a trail version of Telestream Episode or Sorenson Squeeze.
Compressing a heavily compressed file is always going to be a quality hit so do realize what you’re losing for the size reduction.
You’d probably want to use x264 codec CABAC Entropy High Profile for best possible quality .mp4 files if using either of the above.
You might also want to test using Handbrake to see if it can handle your source formats.