To do this, I need to challenge the school's case and, as I said above, I would find it easier to do that with a random LA representative, than the actual person who wrote the report arguing that he can't squeeze my child in!
A random LA rep may know very little about the school, other than the report sent by the head, so may only work to that report. The head may acutally be able to answer questions with authority and knows the school inside out - so they might be more flexible in the flesh than on paper.
Over the years I've heard many (non-grammar) over-sub appeals where the school is represented by a clerk from the LEA. They say the school is full because it's at PAN, and that's that; whereas sometimes if the head has turned up to the appeal, they may admit that actually they can fit one or two more pupils into each year without prejudice. At at least one appeal where both an LEA clerk & head were presenting, the clerk said "the school is full" and the head said "but we can take more pupils". Guess what the panel decided each time.
I've also had heads & senior staff present who've seemed quite scary even to the panel, and others who were really nice to everyone in the room.
Don't forget, they're not against you or your child personally - just against having more pupils in general. Do try to avoid being agressive or confrontational with anyone in the appeals process. It's you against the system, not the individuals; even if being friendly to them doesn't win the appeal, at least it will be less stressful for all involved.