This is unacceptable.
Absence for illness is not unauthorised absence. It is out of your control. There is no way you can commit to improve.
The guidance on school attendance can be found at Working together to improve school attendance (applies from 19 August 2024). This is statutory guidance, which means it is as close to law as it is possible to be without actually being law. The school must follow this unless there is very good reason not to and, even then, must stick as close to the guidance as possible.
Paragraphs 365 and 366 are relevant. They are clear that the school should only request medical evidence where there is genuine and reasonable doubt about the authenticity of the illness and that, if evidence is required, they should not be rigid about the form it takes.
In my view, the school's actions go beyond unreasonable and are unlawful. Given that they are aware of your daughter's medical conditions, the school clearly does not have genuine and reasonable doubt as to the authenticity of your daughter's illness. They are therefore acting unlawfully by continually demanding medical evidence. If they insisted on a doctor's note, that was also unlawful (and, since they have acted unlawfully over this, they should refund any costs you have incurred in getting the medical evidence).
Sickness absence is automatically authorised. The school cannot fine you if your daughter is absent due to sickness. To do so would be unlawful. In my view, threatening to do so is also unlawful. The only referral they can make to the LA is a sickness return if your daughter is absent for 15 or more days in the school year - these do not have to be consecutive days. This is compulsory and is to enable the LA to make appropriate arrangements for ensuring that your daughter is educated. Given their behaviour, I would put a lot of money on them having failed to do this. Everything in this paragraph is covered by the School Attendance (Pupil Registration) (England) Regulations 2024.
A school can only use a parenting contract for poor attendance where there is excessive unauthorised absence (Anti-social Behaviour Act 2003 section 19). Their attempt to put one in place is unlawful.
Put the whole lot together and, in addition to the breaches I have mentioned above, I think it is likely that the school is breaking the Equality Act. If it were me, I would be furious. Far from thinking of home educating, I would be going in with all guns blazing!