The battery detection system obviously isnt accurate and this is common.
However your power supply method isnt the best test, when batteries go dead they tend to go high resistance, that is the voltage drops more and more with load, so that the battery volts will be high enough to extinguish the led untill the transmitter powers up and drags the voltage down.
Try connecting a 100r potmeter in series with the transmitter from the bench power supply, and rather than vary the bench supply's volts, vary the potmeter and see if the led goes out when the transmitter stops working, I bet it does.