I am making a video where I have basically a piano score. I want to show first a bare bones version of the tune and then gradually add parts with nothing else moving so that as the video plays, it looks like the notes are being added to the original file without other parts jumping around. I have figured out how to fix the vertical space between staves, but there is a little bit of movement horizontally (stretching with the bar lines).
It sounds like what you want is "hidden notes". The spacing is just as if the notes are there, but they aren't visible. You can then reveal them at the appropriate time.
Here is the LilyPond Notation Reference page for Hidden Notes.
The two commands of interest: \hideNotes and \unHideNotes. Here's an example from the above reference page.