This is a curious work, in that it's possibly self-inspired.
It would be easy to take this story out of context and assume Lovecraft was a Four, but reading the majority of his works, Four becomes patently ridiculous. His stories are all fuelled by a Five's cerebral horror at an unshakeable fear that things are not as real as they seem, and that "reality" is fundamentally unreliable in some way. "Cosmic horror" is the calling card of the Five, especially the 5w4, who is characteristically more "lyrical", "creative", "imaginative", or whatever you want to call it. Another example is Stephen King's The Dark Tower series, where the world is falling apart at the seams, causing reality to cease to function reliably (time runs inconsistently, space warps between dimensions, or just plain gobbles people up).
What is interesting is that these themes reflect the personal horror a 5w4 sees in the world around them. For a 5w4, reality does not function reliably, like in The Dark Tower. Madness is the undispellable, looming phantom that threatens a 5w4 constantly. Lovecraft's short stories are like a case study of the idiosyncratic, autistic world of a 5w4.
What's interesting about The Outsider is the self-loathing present. It's not clear whether this comes from within or without the 5w4: there is a lot of demonisation of 5w4s as being "losers" when they are not functional, and "weirdos" when they are, which could lead to poor self-image. From within, it could come from the 5w4 constantly measuring themselves against their peers, and recognising that they're almost like aliens from another world, while judging themselves for not measuring up in terms of X or Y markers of success.
As a side note, I think the Enneagram's notion of a 5w4 refers specifically to a high-scoring schizotype (which does not imply schizotypal p.d. or schizophrenia, it just means "someone more batty than the average bear": read up on the schizotypal personality questionnaire).