From your posts, it sounds like what you really want to do is evangelize, and teaching is how you want to do that. Then you should definitely teach in a Catholic school, period.
For me (a public school music teacher in my 19th year), my first priority was to teach the children with the greatest needs – generational poverty, inner city problems, academically underachieving. Boy, did I get my wish! I couldn’t work with this population if I worked in the Catholic schools. They just aren’t there.
Christ told us “Whoever welcomes one of these little children in my name welcomes me; and whoever welcomes me does not welcome me but the one who sent me” (Mark 9:37). I serve Christ in my young students and their families.
And don’t think for a moment that people won’t know you’re a Catholic! My students and their parents run into me at mass all the time. Two of my school students are in my religious education class on Sunday afternoons. I have kids say to me “You have the same crucifix as me!”
Everyone on staff whose known me for any length of time knows that I’m a Catholic. And everything I do at school is seen by children and adults who are atheists, agnostics, Mormons, ex-Catholics, Jehovah’s Witnesses, Buddhists, every brand of protestant, and fellow Catholics. The children may not know the Source of my love for them, but they know they are loved - it’s up to God how He uses that to draw them closer to His heart. It’s not my job to micro-manage God’s work.
Oh, and remember that even when you teach in a Catholic school, you are there to help the children academically as well as spiritually. If all you care about is evangelization, then you might consider becoming a director of religious education at a parish.
The process of discerning God’s plan for you is a
process. Keep moving forward.