30mV full scale, 12 bit ADC, ~7.5uV LSB. Ideally you want the noise to be about not more than 2LSB p-p or roughly 0.3LSB RMS or 2.5uV RMS In a 100kHz BW it works out at 8nV/rtHz, and ideally not more than 2.5uV p-p at LF (0.1-10Hz range) . Possible, but not that simple, and you would also need a Kelvin connection to the shunt resistor.
Something like INA141 chip set for a gain of 100 might be OK (but would need a negative supply - at least -2.5V), however it is not cheap.
Cheers
Alex