What gives you power to the project while charging the battery is called a load sharing circuit, aka power path circuit, which consists of a P-channel mosfet, a schottky diode, and a resistor. Below is a typical example that charges at 1 amp, but the load can draw more than that. It seems the battery you selected is a protected battery, so you wouldn't need the protection parts shown in the schematic.
As shown, if you really need 5V you will need a boost converter to provide that.
Notice that the battery can be charged even when the load is switched off.
For three amps or more, you will need a beefier schottky diode than the one shown.
Anyway, the circuit shows the basic design for a charger with load sharing.