I read this mission during a work shift, so this may not be as comprehensive as usual, due to the lag between reading it and writing this. I just wanted to make some general, overall observations. (I also obviously couldn't listen to the music at work, so I won't comment on that.)
First, on the relationship between Rashida and Falchion. I don't understand why Rashida has so much enmity towards Falchion. Being upset at him filming the tickling scenes is understandable, and, of course, there was the fight scene in the introductory interlude. However, in this mission, Rashida keeps saying that Falchion is incompetent and arrogant, and as a reader, I don't understand why. As far as I can tell, Rashida hasn't seen any evidence of Falchion being incompetent: their fight was close, as they both had red health bars in the final turn. He also displays functioning knowledge of the crossover's main canons. Sure, there are some things he's unfamiliar with, but all in all, he's doing pretty well for a recent recruit on his first mission, especially in light of what he discovers later.
I have an even harder time understanding Rashida's claims of arrogance on Falchion's part. The audience gets a paragraph of narration from Falchion's perspective during the battle, but Rashida isn't privy to that; from her point of view, Falchion is acting just like a regular, wild Skarmory (putting aside the question of just how self-aware Pokémon are, since that varies by title). I also don't see much pride in this mission; Falchion is acting pretty green, both questioning his partner about unfamiliar things as well as testing what behaviors she's comfortable with him performing in her presence. Some changes need to be made on one side or the other: either add some instances of Falchion behaving in ways that justify Rashida's complaints, or change her dislike to something more concrete that we-the-audience can see in the text.
I like both agent's frequent usage of phrases from their homeworlds in their speech and thought. It makes them feel more like denizens of those places, rather than being a sphinx and a metal bird who just talk like Earth humans. I don't know ancient Egyptian mythology well enough to understand all of the references Rashida makes, but it still leaves the impression of her culture on her character, and also gives further evidence of her role as a deity's champion. Falchion's usage of Pokémon descriptors is quite amusing and creative, too. (Although "right off the Zubat" doesn't quite work for me, since that phrase is referring to a baseball bat.)
I also think the mission has some excellent pacing. Despite the jumble of canons present, with each one, you give the reader a chance to absorb it and see it's effect on what was already happening. It made it very easy to follow. The way you divided the mission into acts helps this as well, as it gives mental stop-points for me to digest it in chunks. Considering how long you've indicated the original fic was, (I'm guessing) you're doing a good job of trimming the parts presented in the mission down to just the bare bones needed to show the plot and demonstrate how bad it is. I am looking forward to seeing the remainder of the mission, and the massive disentangling on the horizon.