Before you restore it and go through all the hassle of transfering all your stuff back try a reset network settings. Settings/general/reset/reset network settings.
If it still wont connect to LTE I'd give your carrier a call to see if there's an outage in your local area or to see if there's any account settings that need to be changed/adjusted on their end.
Oh, also make sure your LTE is enabled on your phone. Settings/general/cellular/ enable LTE ON.