Is it possible to save a video of a song you've written showing the BIAB screen with notes, etc, and the sound being played? My crazy idea is to be able to play commonly played songs off a video and not have the software regenerate it each time.

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You could go the video route, or just Freeze the song so it doesn't regenerate ..
just another option
some frozen BiaB songs may actually Open faster than a video and use less resources .. plus you can still tinker with whether you are focused on chords, or the mix .. etc.
YMMV of course, depending on your setup.

Have you tried this.............