Turn off n-p-n BJT on non-zero voltage
Here is a circuit idea that can translate the 0.2V -> 0.4V signal to an on/off signal for the LED.



The two PNP transistors create a differential pair to act as a comparator. The left side takes in the input signal and the right side compares to a reference of ~0.3V created via a voltage divider across a forward biased diode. The load resistor (R6) on the right leg of the differential pair develops a voltage swing of near 0V to just over 1V which works well to switch the output NPN driver transistor on and off. 

The circuit works in simulation but can likely be optimized to lower over all current draw on the power source. As shown the LED current is 9mA and the full circuit current draw is ~32 to 41mA. A sizeable chunk of that is going to heavily bias the diode.