Something extrange happens. In your tests with the batteries, for each battery first check the real voltage in the input pin. Then the output voltage.
This would be consistent with the output off, so floating. But when the voltage detector is active the resistance between out and GND should be very low, close to zero.
So, I understand that this should be like this:
- with Vin > 3.3V resistance OUT-GND => zero.
- with Vin < 3.3V resistance OUT-GND => very high.
Check if it works this way, the opposite or anything else.