If the local area had been stated in the topic title, it might have attracted the attention of someone with local knowledge.
The problem with a Review is that it's a non-statutory procedure with no set guidelines. Different parts of the country do different things. Frequently it takes place behind closed doors without the parents in attendance. Sometimes the parent is not even involved in any written submission (only the primary school headteacher is).
In so far as one can give general advice, I agree with what Chad has written above - the parent's approach should broadly be the same as for a statutory appeal (evidence of high academic ability and mitigating circumstances).
Have a look at the Q&As E12 (it will need adapting, depending on the admission authority). If this is a review where the parents cannot attend, then I would break my normal advice about brevity - you need to give as much detail as possible if you're not going to be there to expand on the case.
Don't worry about "giving too much away". The review process and the appeal process are separate, and the same people should not be involved. [There might be some foundation/VA schools where things are not kept as rigidly separate as they should be, but certainly the appeal panel should have had no previous involvement in the case.]
You are within your rights to request a review meeting just to ask for the schools/LEA's views and to investigate if an appeal would be worth while.
Even assuming the review process is one where parents are allowed to attend, the people there are all acting on behalf of the admission authority and should not anticipate what an independent appeal panel might think.
The LEA and school advised that only breaches of the Government Admissions / Appeals Code of Practice would be acceptable grounds for an appeal.
This was an outrageous statement. Completely wrong and misleading.
the Ombudsman's office advised that the Codes of Practice were only there as guidelines
Legally this has been so - strictly speaking it has been necessary only to "have regard" to the Codes. The new Appeals Code (which takes effect next year) will have statutory force, but even so it will, I believe, draw a distinction between what 'must' and what 'should' be done.