Multiple shunt resistor INA226
Connect all the shunt resistors in series, and put the INA226 across the ends of the string. The range switches are then outside the current sense path, so their internal resistances won't affect your measurements.

For lower currents the values of the resistors must add up to the shunt values you need, eg. 4A shunt = R5, 400mA shunt = R5 + R4, 40mA shunt = R5 + R4 + R3 etc.

In the partial circuit below I have shown FETs to represent your load switches. Note that the TPS22943 is limited to 40mA maximum current.