Grounding issues Arduino Ethernet Shield makes WS2811 LEDs flicker
I suspect that the signal driving the LED strip does not have enough voltage swing. And that ground noise/resistance is what makes the system sometimes appear to work correctly.

With 12 volt LED strips, the data signal into the WS2811 should swing at least 0.3x to 0.7x 12 volts, or below 3.6 volts (low) to above 8.4 volts (high).

The Arduino Due has output signals from 0 volts to 3.3 volts. 

Which would explain the observation that if ground is connected directly between Due and LED then the Due output is never high enough to signal the LEDs.

I would try using a logic level converter between the Arduino Due output and the LED strip data input.