Hmmm, I am not sure how but here is my take on the whole thing.
First off, the French maintain control of their North American territories (so this means the Louisiana territory, the old North West, and what would become upper and lower Canada remain under French control).
So lets put the POD in the Seven Year's War. Unfortunately, France continues to muddle along and another costly war with the English (sparked again by the American colonies, who have not gone independent yet) drains them of alot of money.
This war keeps the French very chummy with the Native Americans and encourages them to send colonists overseas. These are still mostly men who then intermarry with with the local natives.
Unfortunately for metropolitan France, these wars still prove as ruinous to their finances as was funding the American War of Independence. This eventually leads to an alternative French Revolution (there is no American antecedent, but there is still all the Enlightenment thought that spawned both).
With all these butterflies, things go different. A French Republic does pop up, and it engages in wars with its neighbours as they try to stomp it out. The Republic still goes through all that weird ideas they had before, but that is eventually rejected. Instead of getting a Napoleon expy, they get someone more dedicated to the ideals of Republicanism and seperation of Church and state.
Meanwhile, inspired by the French and fed up with the British, the Americans finally revolt and start a revolution. This keeps the British out of continental politics while they try to bring their wayward colonies to heal. Fortunately for the British (though they wouldn't see it that way as they have no alternative to compare it to) this keeps continental Europeans out of their American rebellion.
The Revolutionary wars finally settle down, with France being more conservative with its conquests. It never moves into Spain and it reserves itself to Republican states in Germany, northern Italy, and Poland (some kind of archduchy of Warsaw without the duchy). They wisely also keep themselves from invading Russia. Britain meanwhile has, after years of warfare, crushed the American rebellions (the Americans feeling rather betrayed by the French who felt they had bigger fish to fry).
This France is the world's pre-eminent power and is heavily influenced by Enlightenment ideals, but with none of the crazy (you know, new calendars and religion worshipping reason...). They really open up immigration to their North American colonies and millions of Europeans from across their sphere of influence flood in. Meanwhile, many republican americans still move across into French territory anyway.
Freedom of religion occurs is the norm throughout French North America (still heavy Catholic influence, especially in the Canada's) but there is now room for religious innovation.
Anyway, during the early 19th century, French North America experiences something similar to the Great Awakening. During this French North American Great Awakening, one of the most influential people is a radical minister descended from both Natives and French protestants. He spreads a version of Christianity that is heavily influenced by Native American traditional practices (a hodge podge of whatever tribe his descendants came from) which gains popular spiritual appeal.
This spreads throughout North America over the following decades and this brand of Christianity becomes a significant minority in the French North American holdings.
Welp
, that is all I have got.