I'm not sure there is an easy answer, being out of school has it's own problems.
Runner Mackencheese if medical needs tuition is not forthcoming threaten Judicial Review. mackencheese provision has to be suitable for the pupil's needs, if home tuition is necessary it must be provided.
Zoo DS1 is the same, especially fireworks - a few locally last night had him on edge. DS2 is at a Halloween party and has another one tomorrow.
Muddling the LA's games won't go down well with SENDIST. The letter is typical vague and woolly LA correspondence rather than admitting fault. I would be compelled to call out their bullshit unlawful practice making it clear they hadn't pulled the wool over my eyes, including DD's needs can not be accurately known or met if they haven't been assessed.
The graduated approach is the assess, plan, do, review cycle of SEN support. It will be slightly different for each pupil. Schools and LAs should not have blanket policies as they are unlawful, but the overarching theme is the same. Many LAs try to unlawfully insist on 2 cycles before applying for an EHCNA. Within the graduated approach many break it down into 3 tiers, similar to the old school action, school action plus, statement of SEN. Wave 1 is universal focusing on Quality Teaching First, Wave 2 is targeted SEN support, and Wave 3 is more specialised support which normally includes outside agencies.