Electronic – RPI with power supply alternating between Li-ion battery pack and DC 5V

You're looking for a power management IC.

Here's an example of one

http://www.linear.com/product/LTC3553-2

it includes a LiIon charger, linear and switching regulators to charge from USB and then switch back and forth between USB and the LiIon battery. Possibly not the part you want, but it's a start.

I know this is an old question, but I've been doing some similar things lately and so I'd like to give you a different perspective. First of all, I fully understand why you're selecting these boards and cells on e-bay: to save money. I'm with you there 100%, and I think you obviously know the risks. There are good buys and bad ones as well as mis-representations, and as long as you understand this, the low prices do justify the experimentation. That said, let me answer directly...

I have some experience with battery management chips and circuits, and what you're suggesting "should" be fine, though you would need a supply that at least can be adjusted to a little higher, like 12.6V. Understand, however, that some inexpensive switchers (like wall adapters) can generate a lot of ripple, which I have found can confuse BMS circuits. If the BMS can't do exactly what it was designed to do, all bets are off, and you may never know. So if you use a switcher, consider adding some extra filtering. Alternately consider a higher voltage with an analog regulator, as it will provide current limiting. Its hard to know what inward current limiting exists on some of those e-bay BMS boards, and Lord knows you won't easily carry on a technical conversation with most of the sellers. But current limiting will give your batteries a bit more longevity, at the cost of a longer charge time, but in any case the current must be known by actual measurement. And be aware that picking a 12V 2A switching supply does not guarantee the battery and BMS board combination won't attempt to draw more, which could damage the supply too.

To remedy that, you can often get away with a higher voltage, even without a regulator, using a simple resistor ballast, and solve a multitude of problems. As long as there is something to limit the current. The reason is that these BMS boards effectively go to open-circuit, thus isolating the battery, once full charge is reached. I've verified this many times with a bench supply set to 16V with a 10 ohm power resistor in series, with using the exact same BMS board you linked! The voltage at the input/output point clamps down to under 12V until the battery reaches full charge, and then springs back to 16V, while the cell combo sits at 12.6. Granted, this is not the fastest charge, so eventually you can go to a lower resistance. Sometimes an auto turn signal lamp works well in this kind of circuit, because the bulb can offer an indication of full charge when it completely goes out. I know that sounds like a kludge, but what I'm saying is the BMS board will give you a lot of latitude for making a charging supply, as long as you're careful and measure whats going on while experimenting.

Finally, while we want to avoid repeating other people's answers here, I will second that you be doubly mindful of safety. When experimenting with inexpensive "e-bay" or "ali-express" Li-ION cells, BMS boards, and home brewed charging systems, always be aware that things can go horribly wrong, causing burns, fires, and more serious injury. Do all initial charging in safe, fireproof environments, with adequate protection to yourself and surroundings.