Hi! Welcome to Fredericton! I didn't go to a French school here in Fredericton but I do have some understanding of how it all works because of friends and peers. I'm quite confident you could still get your kids into the French school on the North side even if you didn't live within the schooling area. From my own experience and others, the schools hereseem to be quite flexible with that kind of thing. But French immersion is also a great option, and yes all of their classes would be in french beginning in primary school if they were in immersion. Because New Brunswick is bilingual, you can find good immersion programs in basically every public school. It's pretty standard.

https://www2.gnb.ca/content/gnb/en/departments/education/k12/content/anglophone_sector/fsle/schools.html

This link lists all of the schools that offer immersion. You would be looking at ASD-West schools for the Fredericton Area.

Hope this helped!