I can’t quite see how you can run an LMS as a mobile app that runs exclusively on the device. Many LMS’s have Mobile apps, but they use WebService API’s to interact with the LMS, as a hosted service.
Have a look at Moodle and the Moodle mobile app as an example. The app itself is basically a ‘dumb terminal’ to interact with the cloud hosted Moodle.
If you are really interested, you can run Moodle locally (I do so on my laptop), so you can get to understand its capabilities. It is all open source and there is a vibrant community of developers. It also has WSAPI’s that you could invoke from your iOS app.
… PS. Sorry - I realise I didn’t even begin to answer your question. I will leave this to smarter folk …