Unfortunately, it does not work that way.
The Bluetooth module has to be set up as a specific device, and stay that way. It cannot be switched to a different type of device without un-pairing it from the host, changing the device type, rebooting the Quadstick, so the bluetooth module gets reinitialized, and then re-pairing it with the host. There are multiple things that need to change. The USB does all that too, but it does it instantly when you see the two red connection status lights blink momentarily while the quadstick reconnects as a different kind of device.
One thing you can do is have the Quadstick in XBox 360 mode over USB while having the mouse and keyboard over Bluetooth. That will work for a lot of games.