You could try something rather basic for names. EX "Enigmatic Empire, Enigmatic Republic, etc" I'm sure other people will have better ideas than myself. With that government at the start in the little explaination window discribing how your alien speicies evolved on X planet, you could add a bit of story "The Orginal Enigmatic Aliens giving the tech and to them" and such. Also I'm not sure this would be in the coding but seeing how this tech is even stuff the Fallen Empires don't exactly have themselves I would imagine They would treat you differently than a regular space empire. How Exactly I'm not sure might depend on the type of Fallen Empire (The Techy Fallen Empire could probley be mad jealous of the tech for example). Hopefully Others will have better ideas than myself XD

I feel like the term "enigmatic" should be instead an adjective for whatever empire type you choose. Like for example, if you choose a democracy it'd come up as "Enigmatic Democracy" or if you choose Imperial it'd come up as "Enigmatic Empire."

I'm still considering what I'd do with the Enigmatics, but the Loop should get names like
"<government_type> of Infinite Cycles" or something.

Speaking of Governments, in my current Loop game, I ran into another Loop based empire. They are on the other side of the Galaxy so its fine, even pretty cool. They hate me so its headed for a showdown to prove who loves the worm the most. There is one one thing however, they don't have a name. Their empire doesn't have one at all. Their race have a name just fine, along with their leaders. Only thing missing is their empire name and adjective.

For Loop empire names I have been thinking something like this:
Democracies: Waiting Realms of X or Adherent Unions of X or Worm States of X
Oligarchies: X Adherent Council or Exalted Loop Conclave or X Loopite Tribunal or Wormite Keepers of X or X Adherent Guardians
Dictatorships: Adherent Authority of X or Adherent Stewardship of X or Waiting State of X
Imperials: X Ascendancy or Waiting Kingdom of X or X Adherent Empire or X Adherent Imperium

Enigmatic Excellency, sure you could use that name for something