If that weather station software is written in a portable way (Python?), it could be actually easier to port it on a board which has proper battery support. Then you could simply configure shutdown on low battery in OS settings, and the only thing you would need to implement in hardware is a reset signal as `rising_edge(PSU_OK) AND not(CPU_RUNNING)` which can be done with a couple of BJTs and an RC timing circuit to detect the edge.