Low-power dual-rail ADC
An instrumentation amplifier would be able to offset the differential signal measured across the shunt. An AD8221 could run from smaller supplies (+/-2.5 volts) so this saves power immediately and its current consumption is about 1mA.

Your input signal is going to be in the tens of millivolts so no problem with the AD8221's limited common mode input range (on a split supply) and the AD8221 output swing is to within 1.2 volts of either supply rail so, from a +/-5 volt rail you can get an output that is +/-3.8 volts but you need an "offset" to bias it to a standard ADC input range of (say) 0V to 2.5 volts.

The reference pin on the AD8221 can be fed from a 1.25 volt reference to centre the offset from the AD8221 at 1.25 volts and make it suitable for a whole load of low power single supply ADCs. The ADC input would need to be protected from the AD8221 producing a negative voltage in case the signal from the shunt was bigger than expected but this is acheived by a simple series resistor.

Typically the AD7988-5 (a 16 bit ADC that I've used recently) consumes less than 1mW at a sample speed of around 100kSps so this would fit the bill but TI have very similar offerings as do LT. It has a maximum input current of over 100mA so a simple series resistor of about 1kohm is not going cause a problem BUT check the data sheets.