Now that HoX/PoX has ended, especially PoX #6, we have some indications on where Spider-Man fits in with how X-Men see things:
spoilers:end of spoilers
As per PoX#6, Post-Humanity sees superheroes and superhumans as precursors to mankind no longer depending on or needing to be tied to environment for the purpose of evolution.
This idea essentially gives a concrete answer as to "Why are mutants hated but other Fantastic Four embraced?" The X-Men are natural products of evolution and so they are tied to the natural organic life of the planet and they see non-mutant superbeings as figures who mark the start of mankind no longer depending on the environment.
So to them Spider-Man, Avengers, Fantastic Four are part of the problems and not the solution. So overall, this means that individually the X-Men might like Spider-Man but they would not really consider him "one of us".
On the other hand, given that Spider-Man has had experiences and stuff dealing with where he fits in the scheme of things. His powers coming from the Spider-Totem and so on, it can be interesting if Peter considers himself or would want to consider being part of the road to Homo novissima. Considering how Xavier uses Cerebro to download personalities and consciences into new husks it could be interesting for Peter to compare that to Doctor Octopus' brainscan stuff and so on.
And the transhumanist ideas that Hickman introduces is probably something you could explore Peter pondering from a scientific perspective. Especially since biology is one of Peter's fields.