After going through several tests today, I can confirm that Live Weather most certainly updates from saved flights. If you want static weather, you have to save a weather profile, which answers my initial question - the data is saved in the 3rd line down: “WeatherPresetFile”.
But as @SvenZ has already alluded to, what is the point? Saving/loading flights is still hopelessly broken and it’s been this way since MFS launched. I thought at the very least I might be able to use this feature to archive some of my favourite/challenging approaches, but even the NAV frequencies, CDI/OBS and all other avionics fail to save… It’s very clear why Asobo haven’t yet made any ‘Landing Challenges’ in IMC conditions!
So in conclusion, if you still want to use this feature (even with all its flaws) to save a challenging visual approach with static Weather, make sure you save your current weather as a new weather profile, then set the profile, save your flight, and the next time you load it, you should have the same static weather you flew in last time. Not an ideal solution, but hey…