I have a C3, and checked and adjusted the tailstock alignment, using the two grubscrews on each side of the base.
I used the same technique, having modified a lever lock Tailstock, not originally for the C3, to align it after machining the faces on the base (as opposed to the factory finish angle grinding!)
If the upper part of the tailstock is aligned and secured in position by locking the two grubscrews against each other, to prevent any further movement.
If the screws are firmly locked, the alignment is unlikely to shift, unless the prismatic bed areas wear.
I am bone idle, and throughout my engineering career was always encouraged to do things the simplest way, but as Michael Gilligan says, you have set your heart on another more complicated method, so it will be an interesting project for you.
Keep us informed as to your progress, please.
Howard