Hello everybody,
the issue with the charger circuit can be solved easily and cheaply without too many compromises 
. Here is the circuit:
Transistor act as a diode here. And it is connected backwards on purpose. Model selected was price driven (0,125€ ~ $0.186) and gate voltage must be lower than the minimum battery voltage (important). This model has an Vg of -1.5V.
By the way, dropout depends on MOS resistance and current consumption. This MOS has a resistance of [email protected] and that translates in a dropout of 52mV@100mA.
If you have any questions regarding how the circuit works feel free to ask.
Toni