I was just think about this today and ended up referring to the "lack of self-awareness" as it was the only thing that felt right.
That being said, if I had to pick something that was a single compound word that would indicate the exact opposite of self-aware, I would go with self-unaware.
In certain cases, you might find 'conceite' to be a good antonym of self-aware. As in "Conceited people generally lack self-awareness". It's not a perfect opposite, but works in some cases.
Philosophically, we are all self aware, some just more so than others (self-awareness is always > 0 for humans). Someone who is literally not self aware (in the literal sense of literally) wouldn't be a functional human being. As such referring to someone as self-unaware would be hyperbolic at best. Perhaps this is why the Internet doesn't have a solid answer?