I can only speak from experience with our LEA, but:
We were informed (by our very supportive, proactive SENCo) that in our LEA (Barnet) the usual application for SA (ie. as per IPSEA's model letter) would in almost all cases be insufficient ie. rejected.
I'm not knocking IPSEA, by the way (fabulous source of support) - just flagging up that some LEAs routinely turn down applications for SA, and only agree to assess where the parents and /or school show substantial evidence for the needs of the child. In the words of our SENCo - 'we almost have to show that we are providing the very highest level of support to the child, even 1:1, in order to get the LEA to agree to provide that required level of support legally through a statement'.
We got a SA agreed by writing a very detailed letter of application, stating in detail (4 pages worth) what we believed our child's SEN to be; what the school were alread supplying in terms of support; and how we thought this was sufficient/insufficient (ie. evidencing what worked and what didn't, in order to substantiate our claims about what sort of support our son needs long term).
In fairness, our DS's school wrote a letter supporting our request.
I would urge you to either:
a) get the school on board. Start quoting the professionals at them. Threaten the LEA with legal action if that does not work
and/or
b) appeal yourselves and seek legal advice. Nothing like a threat of action to get the LEA to pull their finger out.
Good luck.
I am still learning up on this whole process (after years of working with statemented young people, not knowing what their parents had endured to secure this support for them) - but an happy to help or advise where I can.