I was troubleshooting Bluetooth connection today and I thought that this is somehow related to software (PipeWire, bluez, bluetoothd and all that stuff). But no. Apparently Bluetooth barely works when WiFi antenna is disconnected from my ASUS motherboard.
Anyway, this might save a lot of time for someone, so I’m posting it here.
You can probably stick the antenna on the back of the base so it’s not easily visible. Or get a separate one that’s a lot smaller.