Weird thought, what about a Mandarin/Doctor opening? It mitigates Mandarin's speed problems (though not as compared to other Doctor openings), it mitigates the economic costs of Doctor's early Copper trashing, and if necessary you can combo the topdecking ability with Doctor trashing. Getting rid of the Copper is also great for later use of the Mandarin on-buy ability, though that's probably a separate question.
I guess if you buy Mandarin first (say you drew 5 copper), you topdeck 5 copper and draw them next turn, buy a Doctor for 3+3, trashing 3 of the remaining 2 copper and 3 estates, and you're still left with 6 copper and 1 estate or 5 copper and 2 estates. If you buy doctor first, then you could at most trash 3 cards (with a 5 copper + coin token opening), and if you're lucky they're all estates. Then you can buy a Mandarin second turn if you don't draw the doctor. But then you wouldn't draw the doctor again, because the next turn you'd draw 5 copper. I guess you'd move on with a buy and play the doctor next turn hoping to kill 3 copper.
I'm not sure it works as an opening. Though playing a Mandarin (and a copper) to buy a doctor does let you trash the card you put back, that seems kind of slow, and you're really just replacing the card on your deck with a doctor in the discard pile.
Maybe I'm thinking of it incorrectly.
Edit: Fixed count.