For me it’s really as simple as “If they can work in Void passing by like a dream, they can do as much for her as well,” and work in some other dream-related shenanigans or such that FGO already uses plenty to travel between the “real” and “unreal.” More specifically, I don’t think her explicitly bizarre and for-the-fans appearance in the summoning system (to take it seriously for a moment and not see it entirely as the milking machine that it is), is fundamentally incongruous with even a short story.
Like, it’s all very easy to integrate without disrupting the flow of the universe’s order as established in FGO. You canonically have every Servant ever, ergo you have Shiki and also Shiki, who has obviously met Fujinon on top of coming from an “irregular Singularity” herself and literally just like Saber Slashy’s introduction at the start of the event is described in bio as:
[considering] her current existence to simply be a dream.
Just like Shiki, she is like an illusion that lasts only a single night… a being that will disappear if the Incineration of Humanity is not reversed.
So why not? You could scratch out something with you and Shiki and/or Shiki meeting in a dreamscape or something unusual like that, and some… I dunno, throw out something adjacent to what happened in the films here, show off Asagami. It could work, but they didn’t want to hire a writer to do it. And that makes me sad.
(Then there’s the… entirety of BB straight up cheating her and the rest into the system and SERAPH’s canonical resolution. Although, I thought that worked pretty fantastically for how it got established in-universe, but that’s another discussion I don’t want to bother with.)