Ultra low power charge pump
Why not program 2 of the microprocessor outputs to make a couple of out of phase square waves .These could be diode pumped to give 3V pK pK minus diode drops .This DC volts is put in series with your existing Batt pos input of 2 to 3 V.Now you have more than enough to pass it through a low drop linear reg thats specified to output 3 V.Your caps wont be big for 10 microamp.