Actually, I'm surprised that it will light the LED.
You are shorting the power supply to the LED/buzzer when you short V_Out to GND_Out.
This is what you are doing:
There's a sort of glaring problem there. V_out shorted to GND_OUT leaves the buzzer with zero volts to work with.
You need a second power supply for the buzzer/LED.
@jbord39 made suggestion that might help if all you need is a short "bzzt" when the short occurs.
Try it like this:
simulate this circuit – Schematic created using CircuitLab
The capacitor provides a little power to the buzzer when Q1 connects the buzzer to ground. The diode prevents the capacitor from being discharged by the short from V_Out to GND_Out.
You only get a short buzz, but better than nothing.
Whether you can get a longer buzz or not depends on the power supply.
I've added a simulated battery, and you can simulate the circuit to find out the voltage to the buzzer.
Changing R3 will change how much current the battery can supply.
Changing R2 changes the severity of the short.
If R2 is a dead short (0 Ohm) then all you will ever get is a short buzz.
In short, the wimpier your battery the shorter the buzz. If the battery is really beefy, it can supply the short circuit and the buzzer.
To get a continuous buzz regardless of the severity of the short, you will have to power the buzzer from a separate power supply.
You do realize that it will take a pretty hefty current flow to make this thing trigger, right?
It takes over 1 Ampere to make enough of a voltage difference across the resistor to reach the 0.7V needed for the transistor to turn on.
What if I have a charger that malfunctions and does not regulate the
charging current correctly, but has a current limitation of 0.5A.
Could I not then have a constant 0.5A going into my battery for which
the short-circuit protection does not protect for, and that my small
battery would most likely not survive?
Lipo batteries generally have higher rated discharge current than charge current. The PCM current limit is supposed to protect against over-current during discharge. It is not designed to prevent charging at a lower current that might damage the cell.
Charging at a current over the cell's rating probably won't destroy it, but the cycle life will be reduced. The greater danger with Lipos is charging over 4.2V, as this will blow it up! Your PCM cuts at 4.27~4.28V, so it should prevent this catastrophe if the charger malfunctions.
For such a small cell the PCM's over-current rating may be too high to protect it in all circumstances, however the safety risk is also low for a battery this size. The PCM has a separate 'short circuit' protection that should protect against very high current draw (a useful feature, because it's very easy to accidentally short the leads when working with these tiny cells!).
Best Answer
There's even more current flowing through the R1 and R2 than R3.
The R3 could be removed with little change in operation.
If Q1 was changed to a MOSFET (eg BSS138) and the R3 removed the quiescent current would be almost zero. R2 could be shorted out and removed as well.
It is not a very good circuit as it relies upon the Hfe of Q2.