You know how bench supplies always have two positive terminals and two negative terminals, and they're always coupled with big (E on its side shaped) shorting links. Well if you remove the shorts, then, run your two thick wires out to the load and two thin wires back to the sense inputs, the power supply will automatically compensate for the voltage drop in the thicker wires.
It seems there is some time before the quadcopter decides it is dead. Can you control the power supply to raise the voltage as a function of the current (and resistence) so that the voltage at the quadcopter is constant (e.g. not at the power supply).
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