5v power switcher: Uninterruptable power source
There are Battery Back-up system manager ICs that are designed to do these kinds of tasks. On a quick search I found this one, but it does not seem to satisfy your current and voltage drop requirements.

Similar ICs, designed to drive external PFETs, exist that can drive the load and provide more flexibility in satisfying design constraints.

If you want to build your own, the basic solution is to have a couple of PFETs used to switch the sources, a comparator to determine when the supply is below threshold, an inverter (or transistor) to differentially drive both gates, and (perhaps) a couple of diodes to ensure this part of the circuitry is always powered. Something like this:







simulate this circuit &ndash; Schematic created using CircuitLab

You must make sure that the comparator and inverter remain within specs (i.e, the input and output voltages do not exceed maximum ratings wrt power supplies), so some more design details will be needed.

Edit: You might need to connect those PFETS backwards, as the bulk diodes might start conducting when power is reversed (forward conduction should not be a problem and power FETs diodes can normally handle the current. Even make D1 and D2 unnecessary. Use Schottky diodes if concerned.