AFAIK, it is not possible to force via a device setting the update of public apps from App Store. You can only schedule via the MDM daemon the OS update (available in iOS 9 and later). The concerned devices are the supervised DEP-enrolled devices. From MDM Protocol Reference (Software Update section), it is written: "The Software Update commands allow an MDM server to perform software updates. In macOS, a variety of system software can be updated. In iOS, only OS updates are supported. Only supervised DEP-enrolled iOS devices and DEP-managed Mac computers are eligible for software update management."
Using Profile Manager as MDM server, you could not update your app automatically from App Store. Personally, I suggest you use a paid MDM solution that support this feature. One of them requests a software inventory to know the managed app versions when the iOS device connects to the MDM server. Then, force automatically a new app install (an update for App Store) if the app version is not the latest version from App Store.