This is not only about the weather but the save / load functionality has a few issues at the moment, I summarized them here.
You can vote here if you will like to see the Save / Load a Flight (including Flight Plan) - MSFS Flight Planner functionality restored, which should include the saved settings in the .flt.
Or you may like to check this thread as well, which is specifically about what settings should get saved.
Are you saying that if you set weather to custom and load a flight situation (.flt), you are unable to change the weather ? or was the weather set on live and you cannot change the weather settings after loading a saved flight?
I would like to remind, that this thread is in the “discussion” category of the forum -
if you would like to see these annoyances worked on; I highly recommend to vote on specific threads, send bug reports, if they don’t already exist - most issue do seem to get reported, the votes affect the priority of what is fixed.