This app was mentioned in 5 comments, with an average of 1.60 upvotes
I don't use anything like this personally so can't say how the battery will be affected. I just installed this one to test though, and it lets you customise the refresh interval and location detection method, so you should be able to set it up to not drain much battery at all.
https://play.google.com/store/apps/details?id=ru.gelin.android.weather.notification
For better GPS, maybe use mifit. I keep the mifit app for updates and stuff, just hibernate it using greenify. And also for notifications try the alertbridge app. It's emoji conversion is amazing.
I also found this to be very annoying, so I use Weather Notification which is always visible on my lock screen. It's far from an ideal solution, but it gets the job done.
This app seems to get the job done for me.
I use https://play.google.com/store/apps/details?id=ru.gelin.android.weather.notification
It's lightweight and simple.