Excluding non-actors who have been nominated for or won (Dr Haing S. Ngor and Harold Russell) Academy Awards and just looking at performers from non-film media who have been nominated, the best precedent that occurs to me is Dexter Gordon receiving a nomination for 'Round Midnight, in which he played... a jazz musician. That's why I think the best bet would be a wrestler playing a wrestler.
Verne Gagne played a wrestler in a movie called The Wrestler in the 1970s when he was still AWA champion but it would need to be more of a heavy duty thing than that. It would need to be in the 1980s or 1990s; possibly the 2000s. I mentioned Piper and I think he's the best possibility without changing too much; he could've done it right up to the 2000s.
Frankly, though, he should've been nominated for They Live. He should've been recognised for his ass-kicking and (alleged) bubblegum-chewing skills.