Update: I got a replacement MPU-6050 breakout board (Aukru - chinese clone) and it works fine in both my breadboard test setup and in the prototype board (once the bad connection was fixed.) It is running on +5V and doesn't seem to mind. According to the board specs it will accept voltages from 3.3V to 5V, so I guess it has the voltage level shifter circuitry described above. It also runs fine without pull-up resistors on the I2C lines, which many say are a requirement for I2C comms according to what I have been reading. Maybe the Arduino Nano has built-in pull-ups. I haven't been able to find a clear answer on that one.
Many thanks for the advice and suggestions.