Well, you don’t have to go to Catholic school to be raised a Catholic. And I would agree that children should be exposed to Catholicism under their parents’ very close supervision. After all, it is the parents who are the primary educators of their children, not anyone else. So why shouldn’t you be closely involved in his Catholic experience? Rather than worry about sending him to Catholic school, can you teach him the faith at home? There are some great religious ed programs out there (Faith and Life from Ignatius Press is the best one I know of - don’t know how that works when you live out of the US) that just require you to sit down and read through short instructional chapters with your child.
Is she concerned about your son’s safety? That’s a reasonable thing to worry about, although it would be naive to confine her worries to when he is around the institutional church. It would be wise to be very cautious about who has access to your children, no matter who they are. I’m not sure if that is what you mean by being “exposed to institutional Catholicism” but I’d be just fine with agreeing to always provide “close supervision” (not sure what she means exactly by that either). I’m always with my kids when they are in church, and I don’t entrust my children to anyone just because they are a church representative.
What is it specifically that you want to do for your child, that she doesn’t want you to do?