Is there a way to essentially do this?
Unit - Create 1 Auto-Turret for player (Killing player) at (Center of Region Attached to (Dying Unit)) facing (Random angle) degrees (Ignore placement)
I can't seem to find a way to call the Region that's attached to the dying unit.. just the unit that's attached to a specified region.
Any thoughts?
None.

An artist's depiction of an Extended Unit Death
Don't bother with the region. You just need the point of the dying unit, correct? Regions shouldn't be handled if you're just concerned with a specific point.
DeadUnitReplaced
Events
Unit - Any Unit dies
Local Variables
DeathPoint = (Position of (Triggering unit)) <Point>
Conditions
(Unit type of (Triggering unit)) == Auto-Turret
Actions
Unit - Create 1 Auto-Turret for player (Killing player) at DeathPoint facing (Random angle) degrees (Ignore Placement)
If you need an area of the region for some other reason, you can create it dynamically using "Convert Circle to Region" or "Convert Rectangle to Region."