Yes, we are talking about an anecdote from the book, "Surely You're Joking, Mr. Feynman!": Adventures of a Curious Character

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Why would Feynman not have published a full(er) proof?

I presume you mean the idea of physical examples in general comes up in "Surely You're Joking"; I can't find any discussion of P vs. NP (or Fermat's Last Theorem, for that matter) within it.

As for why Feynman didn't publish a full(er) proof: a full(er) proof of what? He didn't have anything near a proof of P != NP or of Fermat's Last Theorem... There was nothing of the sort to publish.