1-- I guess: SVA. The verb die is intransitive,and 'a painful death' sound adverbial to me. I don't think it's the same as 'he sang a song'. There may be a linguistic term for this kind of formation-- I think I may have run across it in the past, but if so, I've forgotten it. 2-- The correct punctuation: What is the sentence pattern of "He died a painful death"?
OK-- I like 'cognate object'. In other, Biblical, context it seems to be an object:
Thou shalt die the deaths of the uncircumcised by the hand of strangers: for I have spoken it, saith the Lord ***. Who can count the dust of Jacob, and the number of the fourth part of Israel? Let me die the death of the righteous, and let my last end be like his!