You can update it with the GT-31 updaters and firmware. You'll end up loosing the bluetooth functionality, but chances are the bluetooth is not used, anyway. I have never heard of firmware specific for the BGT-31.
James, I have two BGT-31 both are using the standard GT31 firmware, I am currently using V1.4(B0803T). I did have issues with the Bluetooth settings (BLUETOOTH and BT SNIFF) disappearing from the settings menu, but by repeating the upgrade several times these magically reappeared! I would be interested to here what happens for you. The installer did not work on Windows 10, I had to drag out my old computer. Good luck...