The tough part is having to be 'representative' of the choice for some other team -- if we are all Celtics fans here, how unbiased and can one expect to be when picking for, say ... the Lakers?
How about, once the order is set, just voting via poll for each spot.
I.E., do one pick at a time and let everyone vote from about 5-10 possible players at that spot.
So if say, Milwaukee gets the #1 pick, we all vote on who we think they will take at #1. Whichever player wins that vote is now off the board.
We then move on to the #2 pick. And so on.