Interesting - it's fine once it's running though. So what would cause dosage issues when screw is empty vs screw is full?
The material enters the screw same way in either condition. So that sort of eliminates any potential issues above the screw, right?
You mentioned most solutions that came to mind.
I might try a colder (like cold) rear barrel temp and profile up to a hot temp to the nozzle
if that doesn't work try the other way around

Tried with high and low back pressure? or try to simulate what the screw would do in fully auto?
block the nozzle (with cold slug in sprue maybe?), use back pressure extrude the screw. I might try pulling screw back and trying to extrude in different stages as well; feed zone, mixing zone.

let us know