Here's what the the Bose A20 manual has to say about the OP's question:
There are several end-plug configurations for the headset. Your headset falls into one of two general categories:
• Battery powered: Operates with battery power only.
• Dual powered: Operates with either aircraft or battery power. No battery needed
when connected to aircraft power. This version is often called Flex powered and
comes with an Auto On function.
So assuming that's correct (Wouldn't be the first time a manual had incorrect info) then if you're plugged into a connection with power connected, it shouldn't make any difference whether you have batteries installed or not, the ANR should work.
So, it the OP's ANR isn't working without a battery installed, that seems to indicate the the power isn't connected to the connections in the OP's airplanes.