Best way to back up home screen layout on Android?
Arrangement of apps whether on home screen or in app drawer is part of app data of launcher app. If your launcher app doesn't offer support for Import/Export of user settings, than try using adb to backup your launcher app's data. Verify the backup and see whether the user settings were indeed backed up not. 

If adb solution is not useful than the only way is to get the device rooted and use a root based backup solution, such as by directly copying user settings from launcher app's data directory, or by using an app based backup solution such as Titanium Backup app.

Note: in all likelihood rooting the device would destroy all of your data.