Driving a LED in the most energy efficient possible way
you could make a small high side buck converter with a low rdson mosfet (if you want you can make it synchronous too to improve the efficiency even more) so that it wakes up just in the short time the led is active. You have to pick a mosfet so that it has the smallest gate capacitance though because it needs to switch in the fastest way possible for two reasons: while charging the capacitor, some energy gets dissipated into heat (very very small heat) and the mosfet while switching becomes for a short time a resistor and i'm sure you don't want resistive components all around the circuit. The buck converter can be made with a fixed frequency and fixed duty cycle to save some energy.