High side power consumption measure reversible circuit
Provided that you have some small current flowing through the sense resistor, you could theoretically add two schottky diodes in parallel to the sense resistor. The diode connected to the higher supply voltage would always take precedence because of the voltage drop caused by the sense resistor. 



EDIT #1

If you want to make things a bit more complicated, and implement some sort of active supply voltage switching, you could try the following or something similar:

How it works:

The lower branch forms a long-tailed pair which measures the voltage across the sense resistor, and based on the polarity of the voltage it turns on the corresponding Bipolar / FET. 

e.g. If $$V_1&gt;V_2\rightarrow V_{out}\approx 0V \rightarrow Q_4 and M_2 are OFF$$
AND
$$\overline{V_{out}}\approx 600mV \rightarrow Q_1 and M_1 are ON$$

Therefore, current is supplied by \$V_1\$.
Ditching a conventional opamp and using discrete elements makes it possible to reduce the current consumed by the differential pair substantially. However, it would also be possible to implement it using opamps.