Wow, color me surprised!
That instrumentation amplifier must have some SERIOUS resolution to be able to detect the voltage across the shunt at lower currents. At 1A, that's only 125
uV.
By the way, how are you determining what is a short circuit vs just a really high current surge from a tall geared setup, especially since you are looking for very fast response times (those burst currents can get really high for a couple ms)?