I am not concerned with measuring the circumference. I am talking about measuring the distance between two arbitrary particles of the disc via a taught tape measure (which should ideally be a geodesic of the quotient manifold geometry . I thought that was clear, since I never mentioned circumference.I'm not quite sure I understand procedure 2, I'd suggest the following simple one for measuring the circumference of the disk.

Make up a large number of short rods. You could replace them with taunt tape measures, as well, or little radar sets. It's simpler with rods, though. For more simplicity, make all of the rods of uniform, length.

Count how many of these little rods you can fit around the circumference of the disk.

Multiply the length of each rods by the number of rods. In the limit as the rods become shorter and shorter, the result is the circumference of the disk.