I would just ignore all the crap this school is telling you and apply for the EHC Needs assessment yourself.
You make the application to your local authority (LA). The LA decides whether or not to carry out an EHCNA. They should let you know whether they are going to or not within 6 weeks of your application. If they decide not to, you have right of appeal to the SEND Tribunal. The majority of this type of appeals are won by parents.
If the LA decides to carry out an EHCNA, they must seek advice from a range of people listed in SEND Regulation 6(1)
www.ipsea.org.uk/what-happens-in-an-ehc-needs-assessment
You can ask the LA to seek advice from anyone within education, health or social care, as long as it is a reasonable request, eg OT, SLT, CAHMS or any other professionals involved with your child.
The LA then decides whether or not to issue an EHC Plan. Again, if they refuse, you have the right to appeal.
If the LA decides to issue an EHC Plan , it is the LA, not the school, that is legally responsible for securing the provision in the plan, and the funding for the plan is the LA's responsibility.
Unfortunately, EHC Plans vary in quality from the rigorous and robustly detailed and specified which perfectly match a child's needs, and the woolly wishy washy ones that are completely unfit for purpose, but those concerns are for much further down the line.
For now, in your shoes, I would be applying for an EHCNA myself, and quite frankly, I would also be looking at other, more suitable and supportive schools, whether they are mainstream, special or independent. The one your child is in at the moment (and you are actually paying for
) sounds spectacularly useless.