Or you could just get wireless headphones
I'm really curious if this is a special function of the AirPods or Apple's W1 bluetooth chip in general. One of my other pairs is an older pair of PowerBeats sports headphones. I would definitely consider replacing them with a new set of BeatsX headphones if that's the case.
Battery life is probably an issue too, but with low power wifi SoCs like the espressif stuff, you can probably get away with similar to Bluetooth using the tech that's around now and doing as much possible in hardware and with interrupts as possible.
(I am very happy with my wireless headphones.)