I have a bunch of old TS27L2 opamps with 20µA current consumption, so I would simply pull one of those. TLC27L2 is a TI equivalent. If you can find them, or some other ultra low power CMOS opamps, that might be the laziest solution. Many CMOS opamps have rail to rail outputs too (but not xx27L2).
Wait, sorry. I don't think this chip would be good for operation down to 3V - its output needs even more positive voltage headroom than LM358.
Of course you can always use external supply for the opamp, if you don't mind that. Or try to pass LM358 ground current through the 1Ω resistor, but this requires a slightly more complex circuit where the opamp works in inverting configuration.
There is also the question what happens when the battery is fully discharged and the opamp continues drawing current from it. Not good for unprotected cells.