So, how can we infer anything about the distances between these places if we are given only the total length of the cable?
For one, It would be a lot longer on FE.
Can you back up this claim with some evidence?
Why do you need a measuring tape when you already know how long the cable is?
Can you prove that the cable is perfectly taut?
Addressing the first question: He/she cannot back up that claim because Flat Earth Theorists have failed to provide an accurate map to compare against. Round Earth Proponents have.
Second, the cable is not taut by design. We (I served on a cable-layer ship in the mid 90's) slacked the cable intentionally. To prevent accidents, we dropped the cable as deep as practical. When we made repairs, we slacked the cable so as not to stress the repaired joint. The slack is measured in feet, not multiples of miles.
Parsifal, if you wish to argue a point you need to establish limits. Provide an accurate and scaled map or remain silent.