OK, I think I have a solution.
The other kingdom cards: Native Village, Haven, Bishop, Black Market, (any).
P1 buys BM and pulls the following cards: Island, King's Court, Thief/Pirate Ship, any +actions card, militia/goons/ghost ship, Chapel. He buys 3 curses and uses KC/Island to set them aside; then he plays village/militia/thief repeatedly.
P2, meanwhile, buys Native Village and 4 curses. He sets aside his 3 estates and 4 curses.
P3 buys 5 Havens, 4 curses, and a Bishop.
P4 is Zero R. Hero, who buys nothing whatsoever.
At the end of P1's thiefing spree, the decks look like this:
P1: KC, Thief, Village, Militia, Chapel, starting deck (he can buy KC by drawing village/militia/BM/3 copper). He buys 8 more curses, leaving one, then uses Chapel to trash all other cards, so his deck is just Chapel.
P2: Native Village, 2 copper (he can't get rid of the last two, because he's always holding them in in his hand, and are thus protected from Thief).
P3: 5 Haven, 4 Curse, Bishop.
P4: 3 estate.
P3 then alternates turns of (draw 4 Haven + Bishop, play 4 haven, drawing 4 curse, play bishop, trashing nothing) with (do nothing). He does this three times, gaining 3 VP. Since each other player has at most 3 cards, they now have 0 cards.
P3 draws 5 Haven, and plays them all, setting aside 4 curses and a Bishop. In his cleanup phase, he draws 0 cards.
On the next turn, P4 buys the final curse, ending the game.
This puzzle was deceptively tricky, Blooki! The gist of it is fairly straightforward, but putting together the mechanics was a lot harder than I expected.