Powering a latching relay through a 5V step-up booster is no reliable
I can see a few of issues with your circuit.


The current pulled through Q2 will need to get larger as Vcc falls. But the current Q2 can produce is limited by R4, so the available current falls with Vcc. That is a conflict. You would be better using a P_MOSFET in this position. Actually for a battery operated object, you should be using MOSFETs in general.
Since Q2 is switching an inductive load you need a fly-back diode from ground to the collector. (The lack of it may have damaged Q2 and or the convertor.)
The capacitors on that up-regulator look small. Adding some bulk capacitance to the 5V would be prudent.


UPDATE  Re Your Comment

You would need this







simulate this circuit &ndash; Schematic created using CircuitLab