Turned out to be a board problem ... the BT sketch works 100% with my genuine Arduino Uno on both hardware and software serials, and software serial fails 100 times out of 100 on my Elgoo board where only the first character is received correctly and the rest are always randomly garbled. Hardware serial works 100% on the Elgoo.
When I add the BT code into the self-balancing code it only works at all at 38400 baud though this still crashes the robot once all the data has been read. At lower speeds it crashes - I think this is because the FIFO buffer on the MPU6050 overflows before it can be emptied, though this is still a working theory. I haven't tried higher speeds, I don't think the HC-06 even supports higher speeds than 38400.
Cheers,
David