And FWIW I just searched for "bluetooth headset recording" and found many results, including already existing devices that are a headset which can record the audio.
See, this is my problem when I start to think about stuff. I always take it from the most complicated angle. I didn't look at it as Bluetooth headphones that record stuff directly.
Also searching for "iphone call recording" on your marketplace of choice like amazon, also shows many results including usb dongles which act as a bluetooth dongle as far as I understand.
I neither own an iPhone nor any of these devices, so I can't tell how usable all these devices are!
Probably a middle man bluetooth device which you would connect via bluetooth from the iPhone, which then connects to a bluetooth headset could add a noticable delay, but I haven't seen such a device (maybe because of that exact reason)
See, this is my problem when I start to think about stuff. I always take it from the most complicated angle. I didn't look at it as Bluetooth headphones that record stuff directly.