Voltage drop is voltage drop. It is a value of resistance proportional to a wires length. This resistance needs to be overcome in some manner.
Changing the current lower w/ a relay, will still not get the voltage drop improved. Hence the viable solution is to raise up the voltage to overcome this.
This is why I fast forwarded to the new problem of scarce availability of higher voltage source than 12VDV voltage once utility power is off.
10% voltage drop @ 120 Volts would be acceptable, whereas 10% voltage drop @ 12 Volts is not. Remember the voltage drop is the same for both.
Only solution I see is to reduce the voltage drop by increasing the size of the conductor...#2 aluminum might work, #2 welding/battery cable is even better, but 200' of copper will be costly.
Bottom line, reduce resistance of conductors or increase the voltage applied, or maybe some of both.