The biggest bit of advice I have is to have a squeaky clean cutting tray to cut down on most of the back flash residue redeposit.
The next best bet would be the air suction tray that @evansd2 designed. That might help to keep the dirty vapor from redepositing on the back of the puzzle.
I know of several companies that cut puzzles upside down, since they have score lines on the backs of the pieces to add details to shaped pieces (called whimsies).