I’m doubtful that it’s the same issue with the firmware as it doesn’t show up as a usb-device when connected via cable. I don’t think you can update the firmware without usb-connection.
Have you tried another usb-cable? What make and model is your bluetooth adapter? Clone adapters have huge amount of problems and USB-cables can break or be only for charging having no leads for data.
Is the controller original or 3rd party / clone?
Edit: I managed to just now notice that you actually wrote that the Bluetooth somewhat works.
Does the controller battery have enough charge?