There are several low quiescent current adjustable output low drop-out linear regulators that meet the criteria given in the question. The parametric search on the web sites of several of the usual LDO manufacturers would yield some options.
For instance, Texas Instruments TPS76201-Q1:
- Adjustable Output Voltage: 0.7 V to 5.5 V
- 27 uA Quiescent Current at 100 mA load
- 1 uA Quiescent Current if the enable pin is driven high
The typical application circuit is simple as well:
At the stated operating conditions, power dissipation in such linear regulators is:
P = (4.1V - 2.05V) * 0.127 mA = 0.26035 mW in a worst-case calculation.
P = (4.1V - 2.05V) * 0.1 mA + (4.1V * 0.027 mA) = 0.3157 mW in a worst-case calculation. (Thanks, markrages, for the correction)
Still well under the 1 mW power budget specified in the question.