I think you have too big a range difference. Basically it is 10,000 to 1 (100 / 0.01). This means that at your transition voltage of about 30mV across the 100 ohm shunt, you will see a change to 0.003mV on the 0.01 Ohm shunt. In order for the circuit not to oscillate, it would need a hysteresis of at least almost 30 mV!! (almost your entire range).
One idea would be maybe to use the output of the INA286 to drive your circuit that turns the FET on and off. When the output gets near the positive rail, turn ON the FET. When it gets near the negative rail, turn OFF the FET. Thus you would keep the output of the current sense circuit in range at all times.
However, with this 10,000 to 1 range your low threshold would have to be below 0.3mV and your upper threshold above 3.0 volts. Possible, but not very practical.
I suggest using at least 3 shunts, or reducing the overall current range you can measure.