I think it's also important to take into account France's colonial policies in OTL when considering a potential French Northern Bay Company. In OTL it seems to me that the French colonial efforts weren't based so much on settling the New World (Though Colbert made valiant strides to that end) but rather making money.
If the French were in possession of the immensely profitable Fur country around Hudson's bay, I'd wager they'd be willing to part with less profitable colonies such as Acadia, and perhaps even Quebec if it meant holding on to the Fur country. Much like, in the aftermath of the 7 years war, when given a choice between New France and her Caribbean colonies, France chose the profitable sugar producing Caribbean colonies over the money pit that was New France.
Perhaps in TTL we see a different French focus in North America, where Quebec is abandoned in favor of the North Bay Company with France having colonies in modern day Labrador and Newfoundland (Northern Coast perhaps) in order to keep trade routes open? The British meanwhile dominate the continent controlling all of their OTL lands plus the St. Lawrence, Ohio, and Mississippi river valleys. Content to leave the French holdings in the North and the Caribbean alone.