While I haven't personally tried it (and have since given up on BT audio in my setup), there seems to be a way to extract the pairing key from Windows and have the Linux Bluetooth stack use it (see e.g. [0]), effectively making the Windows and Linux host appear identical to the paired device.
I can attest that this works. I did this for my headphones and keyboard/mouse until I plugged the keyboard into my work mac with the USB cable to charge. I didn’t know that Apple sees this and creates the connection via Bluetooth (Yes used a spare Magic Keyboard). I had no desire to to the whole setup again and now have a cable keyboard and a KVM setup.
[0]: https://brokkr.net/2015/09/26/bluetooth-dual-booting-sharing...