A bit opposite of Rod's reply, I see the Quest 2 is possible to use BT... I would expect this issue to be with the bti-029 adapter and/or its driver or you're not putting the Quest 2 BT into pairing mode after the other pairing.
You might need a different BT adapter or the device cannot be paired the way you're wanting.
Many current BT devices can only be paired to one thing at a time so if you're trying to do two at once then that will probably fail.
It is basically the same thing, I understand that the Quest VR is connected (wired) to the Bluetooth transmitter and trying to connect to the PC to transmit the audio. But on transmitter mode, it only looks for Bluetooth audio devices to pair, and the computer does not show as a Bluetooth audio device. Basically is what you said, "the device cannot be paired the way you're wanting".
Adding a receiver to the setup (connected to the PC audio input), allows the pairing and the audio to be transmitted to the PC