I very much suspect recording during playback is impossible, so that's not likely to happen. Recording needs RAM to temporarily store the recorded audio before it's encoded to the selected format and written to disk, and audio playback already fills that RAM with buffered audio.
I don't think recording while using a plugin could work either...
Anyway, on the original topic, I believe some audio accessory manufacturers make headphones with a built-in microphone specifically for that purpose, it may be easier to buy a set of those... [To anyone who might know] Can the supported DAPs even pass through audio data picked up by the microphone to the headphones..? I'm pretty sure it passes audio from line-in sources (since you can't always hear the audio from the line-in source otherwise), but I don't know if it would do that for the mic since it wouldn't usually have to...