On BT Passthrough mode, the emulated game is communicating with the Wii BT Adapter directly (just like on a real console). Since it's a communication occurring at a lower level, I'm not sure if Dolphin can provide useful logs. Try enabling the log and see if there's any message about libusb...
Edit: ah, I just remembered an inconsistent NAND is also prone to cause issues with BT Passthrough. Try deleting your SYSCONF file (it's located on User\Wii\shared2\sys folder -- location of the Dolphin user folder will vary depending of the OS) and try syncing again. If that still doesn't work, on Dolphin, go to Tools => Manage NAND => Check NAND... and fix any error it finds, then try again...
Edit: ah, I just remembered an inconsistent NAND is also prone to cause issues with BT Passthrough. Try deleting your SYSCONF file (it's located on User\Wii\shared2\sys folder -- location of the Dolphin user folder will vary depending of the OS) and try syncing again. If that still doesn't work, on Dolphin, go to Tools => Manage NAND => Check NAND... and fix any error it finds, then try again...
Avell A70 MOB: Core i7-11800H, GeForce RTX 3060, 16 GB DDR4-3200, Windows 11 (Insider Preview)
ASRock Z97M OC Formula: Pentium G3258, GeForce GT 440, 16 GB DDR3-1600, Windows 10 (22H2)
ASRock Z97M OC Formula: Pentium G3258, GeForce GT 440, 16 GB DDR3-1600, Windows 10 (22H2)