Yes, systematization is better since it is a fictional religion, if I understand correctly.
Your new translation seems fine. You might want to say "opening scene" in case someone reading the English thought you meant "the opening of the alleyway" (not likely but...) If you want to get away from the Japanese structure (which often comes out as static and passive when translated literally) you could say "In the opening scene, Cheryl disappeared into the depths of the alleyway because Alessa's nightmare encroached into Harry's dream."
[Actually, thinking about it a bit more... They have deliberately reversed the usual cause -> effect order of Japanese for emphasis, so even though English is already effect -> cause, it makes sense to use an emphasizer like "the reason Cheryl disappears into the alleyway in the opening scene is that Alessa's nightmare encroached into Harry's dream.]
For the earlier one, now there is a bit more context, maybe: "This was because it turned into a kind of "new religion" as [we] systematized the doctrines in the course of [our] research into theology and demonology."
Probably the first sentence was emphasizing the "unique system" aspect rather than the "for the sake of convenience aspect", so although I haven't seen the original, I wonder if it was more like:
"Concepts and terms for the religious cult to which Dahlia belongs were appropriated from other religious terminology for the sake of convenience, but it has a unique system of doctrines that is wholly separate from any that exist in reality, as previously mentioned."
That would make more sense with the following sentence, but of course it is a total guess on my part.