Further to my reply (I had to run earlier, so I saved the reply prematurely):
There are few ship passenger records from New Netherland in the 1600s. Having a record of the arrival of a particular ancestor is a piece of good luck, and lack of a record proves nothing.
New Netherland settlers came from various different parts of Europe. I think the name Bresie and the other names that have been attributed to this family in old records "look French," and others seem to have reached the same conclusion. Additionally, given the historical context (Huguenots fleeing France to escape persecution) and the presence of some identifiable Huguenots in pre-1664 New Netherland, with more Huguenots arriving later, it is very reasonable to guess that Christopher Bresie was a Huguenot. The fact that he is not the Huguenot Society list does not necessarily mean he was not a Huguenot; rather, it is likely that nobody has found yet affirmative evidence for that hypothesis, and perhaps that nobody named Bresie (or a variant of the name) has ever tried to join the Huguenot Society.
If I were you, I would tell your children that your Brazee ancestors probably were French Protestants escaping persecution, but they have not been traced back to Europe. The lore about Brissac Chateau, Duke of Brissac, etc., on the Bresie-5 profile has to be regarded as mythology.