Voltages are always measured as a differential between two points. That is why your DMM has two wires. If you change the switch position, your loose wire end does not have any sort of reference. Apart from that, the switching itself is going to put it's own signal onto the wire.
The normal way for time domain reflecto metering is to have both the signal generator and the meter connected at all time, and then just figure out a way to ignore the initial pulse. In your case, a comparator that triggers at around 3V should work.
https://en.wikipedia.org/wiki/Time-domain_reflectometer