So. This is my the DAMP's exhaustingive analysis of the current state of Multiverse Theory. It's where the Rose/icosahedron structure comes in. There have never been 20 known multiverses, though you're right, someone suggested it could be a dodecahedron and thus have only 12; that would mean we've seen most of them.
I can see three ways for the HPified PPC to slot into the Rose:
1/ As part of the 'Splinterverse' (see the end of the document). This is described as 'a multiverse where each canon or narrow genre has its own PPC analogue, and where the boundaries between canons in similar genres are weak'. It already has Pirates of the Caribbean, Doctor Who, Star Wars, and magical girl anime 'PPC's; it could easily accomodate a Harry Potter version.
-The flaw with this is that it doesn't account for the fact that these agents are canon characters, not just HP-ified agents.
2/ Your suggestion: a volatile multiverse where everything is in flux. The boundaries between canons would be fluid and vague, meaning a) most fanfic would be crossovers, and b) 'crossovers' wouldn't be a thing, because they're the norm. It would mean that the original canons were mostly crossovers - to use my preferred example, there would probably be a TV show called Headmaster Who, about the nameless, time-travelling headmaster of Hogwarts... sounds like fun! I bet figuring out what 'canon' is would be murder, though.
3/ What I originally meant: a hybrid universe. This would involve some form of Abstract Canonical Entanglement, and possible interaction with the HITeG Manifold around HQ. Check out the Technical Note on page three of the document - the one about the Compromise Theory. Essentially, in some way, HQ (viewed as a separate universe) and the HP-verse would become ACEntangled, creating a 'verse with characteristics of both.
... basically this is a long-winded way of saying 'it's a crossover'. ;)
I think one of the first two options is the most interesting, though. So... why not both?
4/ The Volatile Multiverse. You thought that taking a single look at a multiverse was enough to judge its character.
You thought wrong.
Once, the agents of the Pirates of the Plot Continuum sailed their purloined ships in pursuit of Suvians - then cased those ships in metal hulls, and took to the stars. The Fandom Protection Fleet plied the spaceways, then fell, nearly destroyed by internal feuding. The survivors took refuge underground, developing new abilities as the Multiversal Intervention Agency. They called themselves Time Lords - until the time came that they simply called themselves wizards, instead. Now, in the rambling complex some call HQ, others Hogwarts, they continue their crusade - until the next metacanonical flux...
This version loses the 'switch between them at will' aspect, but absolutely allows them to repeatedly switch to different versions. (The summary above is purely hypothetical, of course - no doubt there were other versions in there too.) Hmm... I may have to get Norlossë to look into this. Given that we've got a Lovecraftean Multiverse out there somewhere, I'm getting a feeling there's distinct planes across the Rose...
hS