The headset doesn't have ANR, but it does have a box which a phone can be plugged into or listen to music. This requires a small watch battery to power.
Just for fun, try changing that battery. If it gets low, some of the internal logic might get a bit wonky and couple some noise into the mic circuit.
I dug up the manual on this unit. There appears to be a few DIP switches inside the control box. The manual wasn't too clear on the setting of the 'power' switch (turn off when directed to by a technician). That might be worth investigating. I suspect it selects an external (cockpit) mic power source on or off. But I'm not certain.
Additionally the cabling of the headset looks simple, so I doubt it has any shielding its self. Is it possible to add some sort of wrap?
Not easily. The critical part of shielding is how it is connected to the comms panel ground/shielding. Not something easily done without major hardware hacking with tiny little wires.
I do notice a quiet background noise when I connect the headset through the cockpit speakers, which stops when I pull the headset out.
I suspect that there may be some sort of interconnection between the headphone circuit shield or common and the mic shield or common that involves the adapter and/or the wiring inside your headset. Noise picked up on the headphone side gets coupled to the mic input, which is much more sensitive. What happens when you plug the headphone jack into the adapter but leave the mic unplugged (not useful for operation, but it could be of diagnostic value). An electronics tech with an ohmmeter could check out your headset and adapters and identify any possible interconnections (or broken shield paths) inside. Someone at Airbus customer engineering should be able to provide a technical level specification of what should/should not be interconnected.