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.
If you want to stick with using the RPi, using an Arduino for power management is not a bad idea at all. It will let you change all your parameters (battery voltage threshold, shundown/reset timings) easily, which won't be the case if you implement this in hardware.