Difficult, this! I think you are doing the right thing by consulting each church school direct about their intake policy. You really need to go straight to the horse's mouth to find out if non christened children from non christian church-going families stand any chance of getting into any of the schools.
If a christening is the only way, you can look at this from two angles:
You are giving your child the option of beginning life as a christian (they may choose to reject this later).
Or, the christening is the unequivocal start of their life as a christian.
Obviously if you see it from the former viewpoint, a christening, IMO, is not such a hypocritical action if you are a non-believer. I really hope this doesn't offend committed christians.
Just for the record, my son attends a church school. He is not christened - neither is his brother, who may hopefully attend the school in a year or two. The school did not ask if my son was christened. However, part of the criteria was regular church attendance. We were doing this before we applied for the school - but not weekly. I have christian beliefs, but do not call myself religious. I wanted my sons to be involved in church life because of its sense of community and fellowship. If, later on, they decide to take it further, that's up to them. I will have pointed them in the right direction. Surely no bad thing. I went to sunday school till I was 14 years old, attended church youth clubs, and went to a church primary school. The fellowship I experienced definitely enriched my childhood in lots of ways. I was influenced and befriended by some wonderful christian adults.
When we filled in the application form for the school, I deliberately put sense of community, not desire for a christian education as the reason I wanted my son to attend a church school. I felt the latter for us would have been too hypocritical. The head did not ask us if we were religious, only if we did not mind our son attending christian church services etc. We were told, after being offered a place, that we were within our rights to refuse our son taking part in the church side of things.
I'm sure all church schools have different criteria so don't take this as typical.
I hope you get a clearer picture of what your local church schools demand. Then you will be able to determine if you would feel morally comfortable about your child attending one.
HTH