The LM4040 is a very stable device. I see no reason you shouldn't use it.
Can you tell us exactly how it is connected?
- which Arduino pins? Is the wire from the LM4040 common connected directly to the Arduino pin or does it go through the plug in breadboard?
- Wire lengths.
It appears the voltage change is driving the current change.
Should we assume the current comes from the 12V battery? How is the current generated?
What else is connected to your circuit?
What is the function of the USB cable in your test?