You can do audio over Bluetooth Classic or BLE. However I suppose the Classic version is more 'usual'.
Also I'm assuming that you have a battery powered headset, but are you expecting the STM32/Arduino to be battery powered? This might make a difference whether you choose Classic vs BLE.
Also if the headset can handle BLE (it should but you need to check).
Also some BLE devices can connect using the classic protocols but better check.
Also the STM23 vs Arduino comes down to what else to you want to do. The Arduino might be a bit more limited but if the Bluetooth connection is all that it has to handle then no problem. If you want to do some of the audio processing you mention, that might favour the STM32.
Unfortunately these are some of the design decisions that only you can make.
Just be clear what you are trying to do, then check all of the components (headphones included) that the can do their part and how.
Susan