Question Integrating wireless headphones with PC and stereo set - using external Bluetooth transmitter

So long after this his An universal wireless headphones / earplugs - a non existing audio hardware?, I have finally bough Bose QuietComfort 35 II Wireless Headphones. Unfortunately as it turns out, my Windows 10 does not recognize that my Gigabyte X570 Aorus Ultra rev. 1.2 motherboard has an integrated Bluetooth. Long story short: I do not intend to spend any more time trying to convince my OS, that I have a built in Bluetooth or to once again correspond with that laughable customer support of Gigabyte. Instead of that I will buy a new external Bluetooth transmitter of some sort, because my very old and almost-never-used one has no problem with showing up in that Windows and paring with these headphones - but produces audio of extremely low quality. [I am making an educated guess here: it is on the account of this archaic device working in standard 1.x or 2.x thus compressing data, because when connected through a Bose wire the audio in headphones sounds good]

And so until now my setup consisted of external Sound Blaster X-Fi Surround 5.1 Pro, plugged on one end to the motherboard and on the other to my stereo Panasonic SA-PM33 CD Stereo System. And I used to have in my audio and video player [i.e. Winamp and Media Player Classic] Sound Blaster chosen as an output device- but now I changed it to just default / primary system [thus to Sound Blaster as it is set in the OS as the main one]

But now The Question arises: should I use a Bluetooth USB-dongle connected to motherboard - or a Bluetooth adapter that uses mini-jack?

And in what standard and with what functions should that external device operate, so that I would not loose audio quality? For now I only know that it should have the frequency range of 20-20000 kHz

I would rather have a jack device. But is my stereo sending electricity through it to, to power the adapter? Or will the adapter also have to be plugged to a power source - or have a built in battery?

I went to some price engines trying to research the marker but as it turns out, jack Bluetooth adapters are a tiny minority and vendors rarely put info about the version in titles / links, thus figuring this out is not so easy