Poor you girls, don't feel guilty and don't feel like a failure, just explore all avenues.
For you damon, in my borough i know that you don't really stand a chance of getting a church school place unless you have a vicar's letter, fake or not fake.
Even if you are at the top of your cofe school's waiting list, and decide to put that as your first choice, beware that any newcomer who has a vicar's letter will get your place, so i would only put that as your first choice if you were: at the top of the waiting list and started going to church.
I suspect in your borough there might not be so much movement of families like in my borough, and not even sure it is really private school land, after all people move to your borough for the good state schools, not sure people there are wealthy enough to afford private schools.
So I would put as my first choice a school where she can realistically get a place, and then do what it takes to improve situation (ie sell a kidney, church, or whatever ).
The worse case scenario is that she gets a place at a shit school, she might have to go there for a term or so, till you can move her. At her age as long as you stay positive, she will not know it's a shit school, she will make friends, etc. Lots of people i know have gone through this in my borough. Priority is not to stress dd.
Do you also have a RC option? Definetely worth pursuing though, forse una visita alla Chiesa Italiana?