If you get a nice steady 5 volt signal on the red wire with sender removed using the red and black harness ground wire to the sender to measure from, then you need to test the white wire and red wires with the sender connected using the harness black wire as your ground reference for each reading. To do that most likely you will have to open the harness and probe the wires nothing that can't be fixed easily enough with electrical tape then harness tape.
You need to ask the manufacturer what should be seen on the white sender wire at proper initialization, or measure it yourself with that test setup on the sender harness wiring in both working and not working conditions and note the differences if any.
You also should ask them what they mean by "grounding problems" that could cause the sensor error you are getting as noted in their directions. If the black wire from the gauge to the sender is supposed to be isolated from the battery/chassis car ground completely, it can be measured with an ohm meter, all power OFF to the gauge and measure on the ohms scale between the battery ground and the sender black wire.
Is the sender metal? If so perhaps making sure it isn't grounded by anything other than the harness.
Assuming the main ground and power to the gauge have been isolated as suggested last week.....