My bengal cat constantly pooped next to the litter box for years.
A lot of the things you listed helped.
You might want to try moving the litter box to a completely new area and then keeping it pristine from there (no existing scent on the surround floor/carpet area). Might want to add litter-attract initially. My bengal also required a huge litter box to feel comfortable.
I found that keeping the litter level quite high helped as well.
It's also possible that the litter you use doesn't clump well enough, so a lot of the litter is dirtier than it looks.
If he does soil the surrounding area, used an enzyme based cleaner to clean it entirely.
If you are willing to lock the cat in one room for a few days/weeks, it could solve the problem. With a small area, the cat will not want to soil most of its living area. It will consciously try to keep a larger area clean.
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