Have you ever seen thread like that?

[Toolmaker51]

Creative idea; if handled carefully. Bobble reversing and bottom out on the jaws, things will go south quickly.
I'd make your die holder floating like common on turret lathes, and drill a port from top to center line to dispense cutting oil. The floating holder would use studs and nuts instead; WAY easier to remove than a sheared screw. Adapt a tap holder, this should work just as well.
Only issue harder to quantify is the footprint, not rectangular like a tailstock or most lathe carriages, making axial alignment difficult.