I'm assuming this feedback is not an intentional sidetone made by your equipment, but instead leakage from headphones to microphone. If it is intentionally put there by your equipment, the best solution will be to modify your equipment's settings.
Better muffed/isolated headphones are going to be the least irritating solution to getting rid of feedback. Noise-cancelling only cares about reducing noise to your ears, but what the rest of the world (ie your microphone) does. If there is not enough padding, sound will escape and get to your mic easily, mostly unaffected by the active noise-cancellation.
- Can you improve the padding/isolation of your current headphones?
- Would you consider obtaining and trying different headphones?
Without knowing exactly what headphones you are currently using, I'm going to lean in the direction of "they are made and marketed by BOSE" and hence recommend that you consider other options. BOSE is a company known for its brand name and advertising. In practice their products are renowned for being absolutely sucktastic and poorly-engineered,
for the price you pay for them. None the less anything with better sound isolation will help.
If you want to pursue your original plan:
- The power microphones provide is definitely not enough to open/close a relay.
- A passive-approach may be possibly using a single FET and a resistor (variable-muting of your headphones) but this will cut out a portion of the signal constantly due to ambient noise
- And active approach with a FET/relay, a voltage-following op-amp, a demodulator, a comparator and a tuning pot would provide a variable 'cut-out' level for your heaphones based on microphone amplitude.
Keep in mind however that this will:
- Need tuning to determine the threshold between "I'm talking" and "I'm not talking", which will vary depending on ambient noise/how loud you think you need to shout at the present moment for the mic to notice.
- Add another failure point to your radio. I'm assuming this is a primary communications method on your craft?