Charging a battery without overloading the rest of the circuit
No, this won't work. For proper charging the battery voltage will probably be as high as 4.2V, which will damage your circuit. Even at lower voltages if the battery should fail as you imagine there would be a very large current flowing through the zener diode, so the diode would quickly fail and then your circuit would be destroyed.

You will almost certainly need some kind of dc-dc converter between the battery and its load.