Well, I finished watching the first nine episodes, and they really gave me a fair bit of an idea to go with what I'm trying to do. As for the rest of the series, I'll make sure to watch and catch up as soon as I can, but I think I have a fair idea of what I want to do with that as a crossover.
Okay, so... the idea goes that Shirou is attending class one day, when a sudden incursion by the Jyamato happens at Homurahara Academy. The Moonlit World doesn't have any record of these beings because of the mass memory eraser that happens whenever the world gets reset, which conveniently prevents anyone outside of the Desire Grand Prix from doing anything to stop them any other way. With the Jyamato attacking Homurahara (and Fuyuki as a whole), Shirou has to make a stand to protect as many people as he can from the monsters, which ends up becoming a moot point when Kamen Rider Geats (no, not Ukiyo Ace from the show, a different person this time around) shows up and defeats all the monsters in the city and destroys the Final Boss-class enemy that would have killed the founder of the Grail Works LTD. This person, oddly enough, turns out to be Hisau Maiya, having survived her fate from Fate/Zero and gotten her desire to bring back Emiya Kiritsugu and Irisviel... while inserting herself into their relationship by accident... and with Shirou and Ilya being true siblings in this reset world.
Problem is, Shirou ends up becoming the new Geats
and the Master of Saber in the Fifth Fuyuki Holy Grail War for the purposes of the reset world. Fortunately, he summons Arturia as she's supposed to be in the Grail Works.
Unfortunately, the Grail is corrupted
again and has the potential for summoning extra Servants. As such, Shirou has to face off against his Counter Guardian self again, while
also taking care of when the Jyamato show up. He wins the Grail War, though he does find it odd that there's another entity inside the Grail, and it's not sharing space with Angra Mainyu. Instead, the entity is Nyarlathotep, who has no idea how come it got shunted over to here instead of where Philemon sealed him away at... and Nyarlathotep is more willing to grant Shirou's wish of stopping the Grail Wars to reform the Grail Works LTD.
Then Shirou wins the Desire Grand Prix and becomes a Desire Deity, which gives Shirou the ability to restore the employees of the Grail Works LTD to their true ability and memories both prior to and after the initial reset that happened because of the original Geats of this story. This means that Ilya now has a physical body to mess with, Irisviel is now the Grail Spirit because of the messing with the timeline, and everyone else in the Emiya household become agents of the Grail Works... along with Rin, Sakura, Medusa and Arturia as they canonically are for this universe. I'm not forgetting your SI, either, nor am I forgetting anyone else... right?
Oh, right, Nyarlathotep is here because of you writing the Persona 5 Royal crossover. That's practically the only reason why I'm including the Outer God and Philemon in this story.