I think mummyloves lucy in the nicest possible way you are meeting problems before they arise 
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You are going to H.E her until a place at school becomes available I presume this is ordinary primary school? my ds2 is ill and when he was in primary we withdrew him to H.E trying to love him better so to speak now I know our reasons to H.E are different but in a place where he felt comfortable in a place where learning could happen at any time and in a place where he could learn in short bursts - learn he did!! he went from being band 2 funded to going into middle sets at senior school is reading level soared his confidence soared and education became 'fun'.
You dont need a tutor you can do this without guidance - look at Lucy and at what her capabilities are ...
letters-can she recognise her letters and sounds? thats her on her way to reading and writing.
Numbers - all day ...counting hunting looking at prices looking at everything
Everything is a learning opportunity and I think that she will benefit a million times by having a period of H.E.
Of course she has SEN but you can adapt things to her requirements- you lack faith in your fabulous parenting abilities.
I remember you won a trip to California??? with mumsnet get some pics of that stick them in a scrapbook and talk about everything ....write things in the book let Lucy copy draw pictures of the dolphins make sounds like the dolphins using plastic bottles with water etc etc YOU CAN DO THIS.
When my son initially needed support and funding wasnt there (school were trying to apply) I as a NNEB went into school did some of the TAs work (putting up backing paper/gluing things/prep work/pe hall equipt) to allow ds to have extra time but if she needs SEN funding will be fought for its not something a parent can pay for.
please have faith in yourself.