So you think it’s the editing system creating a poor encode?
Maybe post a link to the video, so people can see what you are talking about.
If your system is caching the video, the playback will be the same everytime. Are you sure your getting a new uncached version each time? That would be for a short video, a long video is another thing. You really do no offer much info as to length or time in the video the problem happens.
Is there an extream amount of movement of screen change at the time the picture is having a problem?
Have your tried re-rendering the video, and re-uploading the video again? So you can replicate the problem? Maybe clipping the video out where the issue is, so you determine if the source material is causing the problem?
The full res vs 360p could be the way Facebook has thier page code written. I can see them wanting the business material running a better quality, and the timeline footage defaulting to low qaulity. Mainly the low quality is for speed to first frame playback, so you can see the video play instantly when scroll past it. Also having timeline playback set low end reduces your overall bandwidth across hundreds of thousands of users.
I looked into my crystal ball and its not telling me much more than this.