There's a lot of confusion on this thread.
MLL - if your DD has a statement then whatever school the LA sends her to has to be able to meet her needs. You are entitled to name any school on the statement and the LA can only refuse in three situations:
-the school is not suitable for her age or aptitude (e.g. you ask her to be placed at a primary school for secondary stage, or a grammar school when she has SLD, or a special school for severe learning difficulties when she only has mild learning difficulties), or is incompatible with the efficient education of other children (e.g. the child has severe behaviour issues in MS school and these can't be reduced sufficiently by extra support/provision).
-the school cannot meet her needs (e.g. she has sensory integration therapy involving ceiling mounted equipment but the school doesn't have that equipment and it isn't possible to provide it; she uses a wheelchair and self-propels but the school site is huge and essential facilities are based at the top of a hill and can't be moved)
-the cost of the school named by you is not a good use of resources, because they can provide suitable education at a school that is much cheaper.
To get Lucy to a private school, you'd have to convince the LA that you'd have the resources to pay the school fees for that stage of education (5 years) and that the school is suitable for her. Then, you'd have to convince them that it would be no more costly than an alternative placement with 1:1. Lastly, you'd have to convince them that the school would be willing to have a child with her needs and a 1:1.
If you want the LA to fund the school, you need to show that the maintained provisions (the schools the LA runs) are unsuitable for her.