Speaking as a Jew (albeit an unmarried 24yrd old without children) I would strongly caution against sending your child to a specifically Christian institution if you want him to stay within the Jewish faith.
I doubt that Catholic school will "convert" him immediately. However, as he attends the school, his social circle will be built up among almost entirely non-Jews. When he starts dating, it will be with non-Jewish women heavily steeped in the Christian tradition.
I guess the question that you need to ask yourself is: how Jewish do you want your son to be? Leading a true Jewish life is hard, and could require several sacrifices on your part.
The first thing I would do is speak to the Chabad Rabbis about getting your child an Orthodox conversion. It does not matter that you "feel" fully Jewish-- if you don't get him an Orthodox conversion, he will forever be shut out from many aspects of Jewish life. I do know what I'm talking about here-- my father grew up with a Jewish father and Conservative-converted mother. He stuggled with the issue, but eventually decided to Orthodox convert in his 30s. He is very happy he did-- it instantly improved his ability to be accepted within his community. I know its tough, but it's the "price" of admission.
Second, I would seriously consider moving into a Jewish neighborhood. My cousins have totally abandoned Judaism-- despite having Shul-going parents. My brother and I have not. I think a major reason is that my cousins drove 45 minutes each week to synagouge, and went to public school and after-school activities in an exclusively Christian neighborhood. My brother and I grew up in a 30% Jewish neighborhood, with 3 Orthodox synagouges, 2 Conservative syanouges, a JCC, a kosher bakery, bagel and pizza place all within walking distance. None of this should be considered critical of Christians-- I have no doubt that the people my cousins grew up near are genuine, good-hearted wonderful people. However, without the community support, my cousins assimilated-- they didn't turn Christian, they just turned "nothing."
Again, this all depends on how important your Judaism is to you. If you are quite happy with children who will grow up to intermarry, and grandchildren who raise their children as secular Christians, than do as you wish. If however you want your descendents to remain Jewish in perpetuity, you have to take affirmative steps to ensure they are raised in a very Jewish environment.