I can't visualise how the two ends of the wires are located, but is there any way you can arrange it so that the wire is drawn over a pulley or past an idler wheel during the loading process, so that you can measure how long it is by counting revolutions of the wheel?
Some people have used a 1D LIDAR, but that tends to get kind of expensive. You could also have the user pull out a tape measure and have a camera read the number of the tape (Obviosly not with an Arduino, but with a PC or RPi it could be done).
I suspect a sonar sensor would work great. Soft things like clothing or fabric are usually excellent targets at short range (1-5 meters is fairly short). At this distance hard surfaces work well, but the angle needs to be perpendicular to the sensing direction or the sound will reflect off at an angle and miss the sensor. The soft targets will absorb some power, but usually the sensors have enough power to cut through that.
maxbotix.com might be the only option. They are very well supported and documented.