Appeal. As well as I you should appeal B&F. If all DD is currently receiving is home tuition they are lacking too.
Unless your preferred school is wholly independent (is it?) the LA must name your preference unless the LA can prove:
-The setting is unsuitable for the age, ability, aptitude or special educational needs (“SEN”) of the child or young person; or
-The attendance of the child or young person would be incompatible with the provision of efficient education for others; or
-The attendance of the child or young person would be incompatible with the efficient use of resources.
Being full is not defined in law, and on its own being ‘full’ is not enough of a reason to refuse to name your preference (wholly independent schools excepted). The LA has to prove the school is so full admitting DD is incompatible. The bar is higher than many LAs claim. It is more than an “adverse effect”, “impact on” or “prejudicial to”. Unless the school is wholly independent the LA can, and must, name the school regardless of the school’s objections unless they can prove one of the reasons above.
As an aside, EOTAS can include far more than home tutoring. It can include peer support and interaction. For example, it can include music/drama/art/cooking/sports/coding classes, book club, chess club, sport/exercise club/groups, an AP (e.g. a care farm, forest school, outwards bounds centre, tuition centre, art centre), a budget (and TA/LSA/mentor) to access the community (e.g. for things like museums, sensory room, shopping, trampolining, theatre, eating out, travel training…). It can also include therapies (e.g. OT, SALT, rebound therapy, hydrotherapy, hippotherapy, animal assisted therapy, art therapy...) and mentoring (either F2F or online with someone like Mindjam).