How do I invert the output of this current sensor using the other side of the LM358?
Try this topology:



One thing to watch out for, and I don't have time to check on the LM358, is what you're asking for needs the op-amp to be a rail-to-rail type for both the inputs and outputs.

Edit: Here is the suggested circuit simulated:



As you can see, the output ramps down from its high as soon as there is any current flowing. Substituting an ideal op-amp for the LM358 changes the voltages somewhat but not the behavior / curve.

Or have I done something incorrectly?