So as nobody has answered this I will put down what I did, it may not be the best way to do it so bear that in mind!
I opened dev tools and setup a new profile under network speed where the latency was 1000ms and the speed to 50kb/s. (where it says "online" next to "disable cache", click there and go to "Custom > "Add...").
This effectively gives you time to see what is happening on a page as resources load in.
I then went to the "rendering" tab and switched on "Paint Flashing", "Layout Shift Regions" and "Layer borders"
I then just watched the page load with this artificial slow down in place.
It was immediately obvious doing this that I had a side menu loading in via AJAX but when it loaded it was about 1px too wide due to a border on it so it made the whole main section shift to the left a tiny bit.
Impossible to notice with the naked eye but really obvious with the above method. This didn't happen on the automated tests as it was (sort of) a race condition in a browser that depended on load order (using async for scripts as I do other manipulation on the aside) and PSI was consistent.
I hope this helps someone until someone comes up with a better way of identifying the root cause of a Layout Shift.
Update
As pointed out in the comments make sure you test at different screen sizes as well.