Thank you so much for your advice @Imitatingdory and @Aintnosupermum. We are facing some big decisions, and it is frustratingly hard to navigate the options.
We will definitely take the advice and get a private educational psychologist to evaluate DD. (The EHCP sets out some provisions, but they are clearly inadequate.)
I am still not clear on what we would do if the most suitable schools turn out to be two hours away? Seems risky to just move and hope that the LA in that area will play ball. We're currently in Haringey, if it matters.
@Aintnosupermum, I am especially interested in your experiences in the US. The New York area, e.g. Westchester County, is an option for us. I am attracted to something like the Scarsdale School District where it looks like within the same district they have a wide range of options, including Integrated Co-Teaching (ICT), an ASD Nest-like provision which we can't get in the UK.
We are very willing to move - to the US even - if it means that we likely won't have to move again. In Westchester, NY it seems like we can get a broad selection of provisions in one district, which minimizes the risk of having to move again if/when DD's needs change.
In the UK I am still not clear on how to achieve that. Hope somebody can enlighten me. 
Many thanks again, everybody!
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P. S. Thanks also for the list of ASD ARPs. As long as the MS class size is 30, I guess I don't really get the idea. If the child is mostly in the unit (like Netley Primary in Camden), then it seems like they might as well have been in a special school. And if the child is somewhat or mostly in the MS class (like Kentish Town CE Primary), then that's precisely the noisy, high-arousal environment that should be avoided in the first place. Happy if somebody could correct any misunderstandings here. If the ASD ARP children embedded with a class of 12-18 as they do in the ICT programs in the US, then I think it would make sense to me.