Welcome to the forum. I will move your post to the Appeals section, where you should get more help.
Some initial thoughts:
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She only sat entrance exams for the belvedere academy, Liverpool College and the bluecoat.
Did you put any other schools down on the CAF? I'm not sure how many slots you get in Liverpool? For anyone reading this in the future, the advice is always to put at least one school down that you are sure of getting & would be (reasonably) happy with.
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Ou appeal top belvedere did not pass as they have already taken an extra 25 girls in advance of building works to create more space, and the fact that she has attended the prep which used to be the feeder school did not interest the panel one jot.
This would not be relevant to them appeal (I'm not clear how a prep school can be a feeder to a satet school, but it's not relevant anyway). The decision would be based on whether they consider that taking an extra child would be detrimental to the school.
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Our council have allocated her a default school which is rated inadequate by ofsted and I have refused the place.
Again, for future reference our advice is to always accept the offered school and then appeal. The local authority have fulfilled their obligation to offer a school place. If you have refused you are responsible now.
Hopefully our appeals experts can offer you some more advice.