If you hide the weather app's notification on my device, it won't update because it stops running in the background. If I leave the notification present, it will keep updating the weather.
But it shouldn't be necessary, even Gadgetbridge itself works fine in the background while hiding the running notification :)
I really wish it had weather retrieval simply built-in to be honest. I know it's slightly more trustworthy with no internet access in its manifest, but really it's open source, we can check exactly what it's doing.