First, here's the datasheet I'm working from. The latest and greatest from TI itself:
INA193 datasheet
I see the note about the swing range of the output, but there's additional notes just above that and on page 8 and 9 that declare the output bottoms out at 300mV typ. Perhaps the 300mV thing isn't the dominant issue and it really will swing down to 3mV with increased error. It's kinda hard to decide, which all the more inclines me to just empirically test the thing.