All very interesting ....
But back to the question - If the engine is about to run out of oil, as I understand it now it looks like the prop would go into full-fine pitch. That's bad if you shut the engine down due to the high drag, so you'd be best to pull the Stop & Feather while you still have some control over the prop.
That sound reasonable?
I previously thought that the forces on the prop would cause it to coarsen-up as far as it could and I would have pulled the S&F without hesitation, but it's still a valid action no matter which way the prop blades go.