Bluetooth headphone wMic to wired PC audio

I would like to use a Bluetooth headphone for listening to music and also as a headset for skype/google-hangout-call/etc

Attached a graphic showing the setup in mind.

The headphone is Sony wh-1000xm3 which has microphones. Some specs...
Bluetooth Version 4.2
Profiles: A2DP, AVRCP, HFP, HSP
Supported Audio Format(s): SBC, AAC, aptX, aptX HD, LDAC

Interfaced with a PC, which has Line-Out and Mic-Input ports.

While listening to music, I assume that would be over bluetooth using A2DP (which I believe is one way), and then for skype/google-hangout-call it would be HSP (two way) with which mic works but not suitable for listening to music since its lower quality. So Idealy a switch or something to switch between A2DP and HSP.

Is it possible with arduino?
Also wondering if this could work

Haven't been able to find any off the self solution, they are either transmitter or reciever.