Just got to think about it when I read the part of letting the power down, these movements have a stop function in the winding.
If you just let the barrel go in an uncontrolled way you might break the stop finger. I often find these ratteling in the watches from now and then, and I can't see it in your picture of the barrel bridge either.
I don't know, but hopefully, someone here knows what the commonly accepted practice is. I can see that the suggestion to oil the pallet pivots bring doubt to your confidence in the documentation. The only context in which I've heard this being suggested is to combat re-banking by introducing a bit of drag.
Anyway, I'd suggest thin oil, but purely on a gut feeling!