Without a notification bar to show the temperature, even with a weather app running in the background where would you want to see the info?
The best alternative for you might be the Wallpaper Changer HD app. It's SuperBar feature allows the weather, battery percentage, and date to be added to a small bar under the home screens top bar.
It's not headless yet, but the dev has already applied to make it so. "Headless" means running in the background.