Ours doesn't actually it just states number of hours. The tribunal decision did put a ceiling figure in (we costed it out with various scenarios and LA costed it out). The LA over inflated our budget to make it look even more expensive (and in their view outrageous) to tribunal and the tribunal was so unimpressed with LA it went for their (inflated) cost of ABA rather than ours so the first year we actually spent £5k less than we could have done!
Ever since we have had to advise them when fees have gone up - as we have one provider who employs all the tutors thats usually a formal letter informing all parents annually of a fee increase which we forward on to LA.
There is really only one provider here so its not been an issue so far.
So I don't think it has to be in the statement but there does need to be written agreement either between you and LA or a max budget set by Tribunal (only binding to next review).
Our statement does not name a provider, rates or who invoices etc. Mostly ABA providers invoice LA direct - in our case LA was messing us around (suggesting they put ABA out to tender) so we agreed to employ the provider of our choice to get round tender rules (which I was not convinced apply here but anyway) and we invoice the LA and then pay ABA - our invoices are paid within 7 working days by LA and we have 28 days to pay ABA. Its always worked so far. Getting LA to contract with provider direct is probably easier.
I don't understand your EY provider taking off an admin fee! Are you sure thats not for tax, NI, holiday / sick pay etc??? Our tutors are billed to LA at £25 p/hr (but they don't get that, that covers the cost of provider's premises, admin staff, cover staff, in house training etc etc the tutor probably gets a similar amount to yours)
In my experience leaving the school out of it and you or LA employing provider direct works better. Schools are becoming more keen on this than in the past as funding changes mean the first £6000 of any statement has to be paid by school. Here the LA thinks that cannot apply where the statement is Education Otherwise (ABA) and it will continue to pay the ABA costs in full (not ask school for £6k contribution), so it will be treated as an exception. It means also you don't get into wrangles about whether the school has enough money to cover the first £6000 or them trying to share your staff with other children etc. It also means it doesn't cost the school any more money for your child to go (except a bit of IEP, liaison, meeting time). If the school employed the tutors I would think the school might have to pay the first £6k.
Schools used to want to employ the staff / use own TAs but can now see the benefit of private staff paid direct by LA (and not their budget) coming in and working as LSAs.