L.R.
For the two religious schools: I would talk with the administrator at each who handles inquiries from potential students' families. You don't have to say as soon as you walk in the door, "We're not religious"; just be neutral, get all their brocures and paperwork, ask for a copy of the curricula for grades K-3 or so, and listen to the pitch. Do ask "Do you accept children who are not Catholic?" and "Do you accept children who are not Lutheran?" The answer VERY likely is yes; most religious schools (around here at least) have plenty of students who are not of that specific faith. Then ask for a way to contact some parents of current students there or ask if there is any kind of parent advisory committee you can contact. Although those parents will of course be ones who will give "good reviews," you can at least ask them "What if a child's family is simply not religious, as opposed to, say, Christian but not Lutheran?" See how they react.
Be aware that most religious schools will have, as part of the curriculum, some form of a religious education class at some point, if not every year, and some will require all students to attend chapel services weekly (or more or less often). See if students can opt out of any of those things. But these are after all religiously based schools so recognition of the religion is to be expected. So a lot depends on whether you would be OK with your child going to a religious school for the education and also having to attend chapel etc. as part of the deal.
Just a note on that: In Britain, which is a highly secular society where very, very few people attend any form of worship services or identify themselves as religious, the state-run (public) schools do still require "RE" or religious education as part of the public school curriculum. The idea is that one should study religions -- all of them, not just Christianity -- as part of the history and culture of the world. It's not intended to proselytize to kids and convert them (though once that was indeed the case, long ago). If that idea is OK by you, then maybe you would do OK with having your kids at a religiously based school. All depends on what you feel about it.