i just ran across your post. I know only the information you have given me and what I could find on ancestry. You mentioned that Jules was born in Belgium,LaCroix. I wasn't aware of the LaCroix part. I haven't been able to find any Belgium records for this family. I have Jule's father's name as Augustin. The surname might have originally been spelled, "Appelman" or "Appelmans" depending on where the name originated.
Have you had any luck since you posted this query? I have searched so long and hard with little to no result, I'm wondering if I've been searching with inaccurate information. Could Jules father have been born in Holland, as one of the censuses suggest under his brother, Arthur's name?