Firstly, have you requested an expedited hearing on the basis DD is out of school?
Is your preferred placement wholly independent? If not, being full is not enough of a reason on its own to refuse to name your preferred option. The LA has to prove they are so full it is incompatible with the provision of efficient education for others or efficient use of resources. Now there is a point where they can do this, but it is a far higher bar than LAs and many schools admit. It has to be something tangible and specific and is more than an “adverse effect”, “impact on” or “prejudicial to” rather than the vague and woolly excuses LAs often rely on.
Personally, I wouldn’t electively home educate. If you do that, the LA has even less reason to concede the appeal because EHE relieves them of their duty to provide DD’s education.
EOTAS/EOTIS isn’t EHE. The LA retains responsibility. EOTAS/EOTIS via the EHCP is only legally possible if it is inappropriate for provision to be made in a school. Unfortunately, most have to appeal to get a comprehensive EOTAS/EOTIS package.
Alternative Provision comes in many, many forms. It depends on the needs of the child and their circumstances. For example, it may be F2F tuition (at home, in the community, at a tuition centre, at a hospital school… It more be traditional academic tuition or not. It may be child led or not), online schooling, online tuition (1:1, group, pre-recorded…), at a formal AP (e.g. care farm, forest school, outwards bounds, a gaming AP, art, cookery…), mentoring (at home, in the community, at a centre, online…), sports. There are numerous possibilities. What AP is the school offering? (Or rather what AP is the LA saying the school will/should offer? Although the LA is ultimately responsible for education for those unable to attend school/the provision in the EHCP.)
If DD is unable to attend school because of her health/SEN, the absences should be authorised, thus no fines/prosecution.