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
http://www.tinyosshop.com/index.php?route=product/product&product_id=993

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