About Bluetooth Headphone

There is definitely a bluetooth bug in Monterey. You will find references to your problem in other forums, e.g. mac mini. My BLARO headphones worked just fine before my upgrade to Monterey. Now, I get the connect/disconnect/connect/disconnect problem. My other headphones will sometimes not connect and then sometimes they do connect and appear in Sound Preferences. However, they are unusable because I can't switch the sound to the headphones. Yes, I've tried all the "fixes": reset NVRAM/PRAM, delete BT plist file, reboot iMac, remove and then pair device, reset the headphones. Maybe I've missed a "fix".

I am thinking this bug will be with us till the next Monterey update. Fingers crossed.

if it was me then I would make the computer forget the headset in bluetooth

and then put the headset back into pairing mode and connect it from scratch

My money is on some sort of handshake problem, e.g. security pin, passkey.