ANR headsets can only cancel acoustic noise which would otherwise have reached the ears from the environment. They simply cannot reduce the 'transmitted' noise from the other microphones on the system which is fed to them electrically. A headset cannot make the distinction between wanted signal, for example speech, and unwanted signal like the microphone noise.
That you notice the microphone noise so clearly on your ANR headset is a tribute to the acoustic noise cancellation being achieved by the headset. Without the ANR, the mic noise would be masked to the point where it blends into the other ambient acoustic noise.
In your case, possibly the best solution would be to prod the other pilot to switch off his mic and then you ought to be able to enjoy the full benefits of ANR tranquility.
In audiophile circles it is often found that upgrading one component of an audio system gives a worse sound than before; as the saying goes 'The wider the window is open, the more !!!! flies in'. In other words, by upgrading for example the loudspeakers to a more revealing pair, it often allows you to hear the faults of the rest of the system, CD player, amplifier etc, which were thankfully masked by the rubbish speakers before.
To give an extreme example, as you become progressively more deaf with age, even the cheapest, nastiest boom-box stereos start to sound as sweet as the most expensive hi-fi, saving you fortunes of wasted cash!