The only other thing I can think of is to try and DFU the phone. I know it is annoying since you have to restore your backup, but it may save the phone.
How to DFU an iPhone. Make sure the screen stays black when in recovery mode as the directions specify. I would try this, and then set the phone up as new, only to test the Bluetooth.
If it still doesn't work on a new setup with no restore, after a device firmware update through iTunes, this confirms it is hardware issue. If it works, then erase the phone and restore a backup. If it happens after restoring, this means it is a corrupt backup or software issue in your backup. If it still works, then you are good to go!
Give this a try. A pain, but will give you a definite answer. Good luck,