But you could wait to have two exiled players to check that moral loss condition...
If the exiled player does not show his/her objectives, you could have the traitor being exiled, but the other players still exiling another player. If the exiled player objective card is show, this situation cannot arise.
Should one show the full objective card, or is it enough to show the "traitor" mark (or its absence)?
I don't have the game with me, but I believe that the Exiled cards explicitly tell the player whether or not to show their original objective card.