USB Audio Adapter for Bluetooth Headset
Technically, those adapters don't exist.

The USB device should be recognized by the Mac OS not as another Bluetooth device, but as an Audio Device.

I agree with you on this one.  You definitely don't want to have a secondary Bluetooth adapter appearing in macOS.  So to clearly define what you're looking for is a USB audio interface with Bluetooth capability.
We have to make one.  Here's what you need:

USB DAC Audio Interface.  I personally like and use the Behringer UCA202. I have mine attached to my iMac and plugged into a Bose Wave Radio.
Bluetooth Transmitter/Receiver.  I also like and use the TaoTronics Bluetooth Transmitter &amp; Receiver.  I have mine hooked up as a transmitter to an LG TV in the bedroom so I can watch TV using my regular Bluetooth headset (I didn't want one of those proprietary things where I couldn't use the headset of my choice).

Now, all you have to do is connect the USB DAC to the Bluetooth transmitter and then pair your headphones to it and not the Mac's Bluetooth.  (Here's the awesome part) Since it's attached to the Mac as a USB audio device, all you have to do is turn the volume up to the max on the BT transmitter and use the Mac to control the volume on the DAC.  You will maintain full volume control through the Mac.  However, the volume control on the headset will manipulate the volume on the transmitter, not the DAC, so you'll have to keep that in mind.
As far as stereo/mono, this setup will transmit whatever you send over.  For instance, the TV I have mine attached to does stereo and mono output and the BT transmitter handles each with no problem