Hiya star. Can't believe I missed your thread. I had a feeling you were thinking of HE again 
This LA sounds a bit public-sector-dozy but not evil. Same with the school (yes, even though they are indie). Which is novel. The trouble with your request, though, is that you're asking for something that doesn't have a category. No budget allocation, it's no-one's job, mucks up their nice tidy pupil-allocation lists, doesn't fit any stereotypes and probably never been done locally before. And if they try they might end up with multi-agency meetings, arguments about resource funding, bad feeling with the indie school and probably a bol*king from their strategic management team.
Plus your plan rips apart the polite accepted-fiction of the professional and the pen-pusher working productively together for the child's benefit; always knowing best, earning the eternal gratitude of well-meaning but rather stupid parents who simply don't understand special needs, local government, disability, or indeed education of any sort.
Go back, and be very grateful for their carefully considered advice and their agreement to look at all the options for September. Their agreement for this school in the first place, the marvellous progress in non-academics compared to the previous terrible experience in the last borough, anything complimentary you can say without actually lying.
You've taken the panel's comments on board, and discussed them with a number of wise friends whose dc have a variety of SEN
and you think perhaps the LA asking the school to consider flexi-schooling might be a good option for the next term or so
.
Lets them give in, whilst meeting their needs: saving face, staying out of trouble, and feeling there was a point coming to work.