I believe that enigma is the name, as they are part of the same project, but i'm not completely sure.
They're separate projects. LGM is just an IDE, Enigma is just a compiler. Each has their respective team and team leaders.
The 'combined' name is not determined yet, but some suggest it should be ENIGMA, since LGM is just a de-facto standard IDE that happens to be included and can be rebranded, while others suggest it should be LateralGM, since the compiler is never seen - what you actually run is LGM, and the compiler just happens to be one of possibly many plugins. I don't think a combined name is really that important, as people don't really confuse Eclipse/Netbeans for Java.
How transparent and easy is it actually possible to make the combination for the end-user, in regards to set-up and execution?
LGM comes packaged in one jar file, making it very portable and very easy to include in another project. ENIGMA has capitalized on that, and included it with their compiler. As such, all you need to download is Enigma, and LGM comes with it.