You have two separate issues.
The EHCP.
Is it a refusal to assess or refusal to issue?
You need to prove the legal test is met.
Being in receipt of section 19 provision doesn’t change this. An EHCP is important even if section 19 provision is in place because an EHCP can provide more support (e.g. therapies such as physio) than s19 provision would provide.
Also, whilst the LA has the power to provide provision for post 16 pupils, they don’t have a duty to ensure DC unable to attend still receive a full-time education, so most don’t. Therefore, it is important to plan ahead.
Section 19 provision
Section 19 of the Education Act 1996 ultimately places this duty on the LA, not the school.
The LA is responsible for ensuring DD receives a suitable full-time education. Or if full time in any form is not appropriate, as much as is appropriate. 6hrs is not full time.
Provision should be suitable for DD’s age, aptitude, ability and SEN. It should offer a good quality education equivalent to that offered by MS as far as DD’s needs allow (i.e. it shouldn’t just be maths, English and science. It needs to be a broad and balanced curriculum). It should consider DD’s pastoral, social and emotional needs. And as far as possible, DD should be involved in decisions. You cannot be compelled to deliver provision.
So I would be asking for more tuition, more subjects, not just academics, staffing, resources/equipment. What subjects don’t you think are possible outside of school? You would be surprised.
As an aside, if DD wants to attend school and is well enough, the school can’t refuse to allow her to attend unless they are formally suspending her, which you could then challenge.