I remember that on my previous phone running Android 4.3 I was able to disable background usage of mobile internet, yet enable it for specific apps. Instructions to do so can be found on the Google support pages.
- Open the Settings menu on your device.
- Under Wireless & networks, touch Data usage.
- Make sure you're viewing the network for which you want to view or restrict app data usage. If more than one is displayed, you can switch between them by touching the tabs at the top of the screen.
- Scroll down to find the app whose data usage you want to view. The blue bar for each app shows its total data usage for the cycle displayed above the graph near the top of the screen.
- To see more data usage details for an app, touch its name.
- You’ll see a graph showing just this app's data usage for the specified cycle. Below the main graph, a pie chart breaks out foreground versus background data usage for this app.
- "Foreground" data usage means how much data an app uses when you’re using the app.
- "Background" data usages means how much data an app uses when you’re not using the app.
- To stop background data usage for this app, check the box next to "Restrict background data."
- Read the message that appears and touch OK.
This however does not seem to apply at least on the LG G3 running Android 5. Is this something LG moved to some other place? Or has it moved on Android 5?