You are asking a difficult thing.
Currently the official Sony SDK STR encoder is the only working Playstation movie encoder.
I assume the only difference between .STR and .MOV is one has the extra 40 bytes of camera data that is unique to some FF7 movies (unconfirmed). So even if you did manage to properly re-encode the video, you would still have to manually edit the files to copy the camera data.
I hope to eventually add Playstation video encoding to jPSXdec, but with the current schedule that won't be for another year or so. Even if I started working on it right now it would still take me a few months.
If you have the drive to do it yourself, nearly all the information is available to you. The STR file format document explains all the details of Playstation video, and I have outlined how to write a Playstation video encoder
here. Even if you don't have the drive to learn everything, you will still need some understanding of the PSX STR format to ensure your video data is properly formatted.