I'm back but I see senvet and attila have stepped in with good info in the meantime.
I'm afraid the SENCO and LA are spouting nonsense - it may be how they want/think the system should work, but it is not the law. Many applications for statements and now EHCPs get rejected because it is a fantastic delaying tactic for the LA, and, (even better for them) some parents may give up at that hurdle. If you made an application for an EHCP and it was refused, you could appeal to the tribunal who would give short shrift to the LA's unlawful criteria that a child needs 'Tier 3 top up funding' before an application can be made.
So see an 'automatic rejection' of your application as a way to actually get past your LA's unlawful policy, and get the decision from a tribunal who will apply the law. (Though in practice many LAs change their decision not to assess once you take out a tribunal case and they actually have to defend their refusal.)
The LA get 6 weeks to decide on your request for your DC to be assessed for an EHCP so if you did it now (!) there are still 6 weeks til schools break up for summer. I think it is only the long summer break which doesn't 'count' in meeting the 6 week deadline.
Yes, an indie is a private school that charges fees. It sounds as though your DD is in an academy, so not LA controlled but still state funded. Yes, that would class it as 'mainstream', in that it is not a 'special' school specifically for children with SEN. A NAS school is a National Autistic Society school and they are fee paying special schools.
Ds was out of school because of anxiety after his first term of Y7 and at the start of his Y8 (last sep) he started at Interhigh. It worked fairly well for him at that point but he really needs a wider curriculum than it can provide and although he can find it challenging, he also needs the social and academic stimulus of a 'brick' school. Some LAs will fund Interhigh ( as part of an EHCP) and some LAs have their own online learning 'platform' for DC who can't attend 'brick' school. Our LA wouldn't fund Interhigh, but by this time DS wanted to look at going to 'brick' school again.