Using a BJT to enhance battery life
This is fine if you are having an external power supply at the collector of the BJT but in reality you aren't and the current for the base, as you said, would be taken from the battery but so would the current through the collector!

So while the base current is sort of negligible it will still actually be draining more current than originally so there is no point in this being here. And with the voltage drop of the base-emitter junction there may not be enough left for the voltage drops required over the LEDs you have.

While the battery may claim to have a 560mAH, it is not likely that it will actually play out this way, they are quite often conservative with these values and it may actually be higher and last longer than you expect but as mention by Andy, the terminal voltage will droop due to increases in the internal resistance of the battery which again could stop the LEDs lighting up to a satisfactory level.