The name LaChance might be a French spelling for an original name Lasancz or Leschanz or similar. There were families called Lasancz in Hungary, and the current phone directory of Austria lists a number of Leschanz. Perhaps the original name was from the region of today's Slovenia.
Crown prince Francis Ferdinand and his wife were assassinated in June 1914. This assassination triggered WW 1 which started just a month later. There was nothing dishonorable for Francis Ferdinand, therefore I have no idea why his offspring could have been persecuted. Very few people were able to emigrate to North America from Austria-Hungary during WW 1.
You told us about your greatgrandfather (the archduke) and your parents. Who were your grandparents? Where and when were they born? Where did they marry and where did they die? When did your parents arrive in Canada? What was the maiden name of Gladys?