Eidolon would be WAY too confusing in my opinion, what with XIII and all.
I actually kind of like the model thing, it has a really dehumanizing feel to it. I'd go with model or project depending on each specific context. And both could be used in capital as Model or Project when you need to give it a proper name. Again, Model seems to really go well with that. Giving that most of them have numbers, you'd have an implied Model 2Model #2/Model No. 2 type feel. And you could capitalize it on the same basis as you do with soldier vs Soldier so people know you're being specific.
Of course you could always hit up the thesaurus. I'd advise against replica, but that's total bias as a Tales of the Abyss fan. Facsimile is nice and fancy sounding, as well, and also implies an imperfection in relation to the original.
My main votes would be Model or Facsimile. I mainly would like to avoid made-up words though.
PS - I'm pretty sure in Tales of the Abyss even the Japanese used replica (in katakana), so it's possible if that's what was intended they would have probably used it? It's two very different companies though so it may just be a difference in writers.