I feel for you OP, you’re in a tough situation and it must be hard to see the wood for the trees.
Your daughter going to this school is obviously something you really want to happen. You seem to have all sorts of hopes pinned on it and are prepared to risk your whole families well being for it, it must be an amazing school.
Sometimes I’m life you really need to take a good step back and try and look at your situation from the outside. Your daughter is 4, barely more than a baby, she doesn’t care what school she is at. Also at her age a school move won’t affect her and she’ll settle in quickly wherever she goes.
You’re pregnant again, and you say this is a fee paying school. Can you afford the second set of fees when you younger child starts, remembering that your dds fees will likely be considerably higher by then? If not then take your money and run now.
You have your own house, where you could
provide a stable life for your family. Your daughter could have local friends and you could walk her to school. Also looking to the future for you, you could adapt your own house to meet your changing needs. Why would you want to cram yourselves into an unsuitable tiny flat, which wouldn’t even be in the ideal location? You’d be paying through the nose for your family to live in miserable conditions. What about when you have two teenagers in your tiny flat? I can tell you now that my dds would rather have their own space and go to the local comp than have to live in a cramped flat, sharing a room, but have the privilege of attending a fee paying school.
I’m sure the school is great and in your ideal world it’s the education you’d like your children to have. But sometimes we have to just do the best we can with the hand we are dealt. Go back to your house and find a local school your dd will like, forget about this school, be happy, make local friends and enjoy your family. Life doesn’t have to be this stressful.