How to make sure that Bluetooth headphones will not suppress musical instrument live playing?

I am a classical musician (cello, piano) and regularly teach online classes through apps such as Zoom, Teams, and FaceTime. The setup usually is: student on live call through the iPhone, music score on the iPad, and either me at the cello or at the piano. In the 2020-23 I used the Powerbeats Pro 1, then in 2023-24 I switched to the Focal Listen Wireless (PBP had a dead battery), and last week I switched to the new Powerbeats Pro 2. A huge problem I encountered with PBP2 is that they suppress my cello/piano playing so that my students hear only muffled inaudible garbling sounds. If I sing, no suppression is applied. Through ChatGPT interrogation, I learned that this could be due to the beam-forming mics alongside software calibration that considers musical instruments as something to suppress. As far as I knew, this was an option in apps, not a hardware thing: I could set up Zoom to use original audio for music lessons, and auto-suppression for business meetings. Now, even if I do that, the PBP2 take precedence. How can I solve this? The iPhone (iOS in general, I think) doesn’t allow separation of audio input and output—that is, once I have worn the PBP2 they will use their mic for input. Using an external mic is not a guarantee that iOS will let me use it instead of the earbuds mics. Teaching without earphones will cause return feedbacks. Is there any app or setting that allows input/audio separation on iPhone? Is there any modern bluetooth headphones that does not suppress this? Please notice that this effect is active even with ANC off—which is disturbing. I realise the market of classical music teacher teaching online is quite small for software/hardware developers to even care about it, but if you have any suggestion or even just a clear explanation of why this happens, I will greatly appreciate it. Thank you for your time.