A player has exposures that could only be part of one hand on the card. That hand requires a pair and three of that particular tile have been discarded. Should I call that player's hand 'dead'?

Yes, you should declare that player's hand 'dead'. The player then exits the game and no longer picks or discards, which leaves more tiles, and therefore a better chance of declaring Mah Jongg, for the remaining players. Some players feel bad about calling another player's hand 'dead' and prefer not to do so. However, this is just part of the rules of the game.