"The offered school has 382 pupils in total 63% are on free school meals, 34% are SEN. The school is rated inadequate from ofsted."
I'm wondering what you think that means for your DS?
~63% on free school meals is irrelevant.
~34% on SEN register means that they are good at acknowledging and recognising SEN.
~An inadequate OFSTED rating means that the Local Authority will be all over them like a rash, with an injection of expertise and resources, so a rapid improvement is likely to occur.
The bottom line is that if you're lucky, the Ed Psych will recommend your preferred school. But they are professionals and they are obliged to write truthfully, so if they think his needs can be met at any school, they shouldn't be swayed by your wishes.
If your DS is the only appellant, then you might win quite easily. But you have to consider what will happen if 11, 20, 35 people appeal...where would your case sit in terms of strength? If a panel decides the school can only admit an extra 6, for example.
I'm glad you're going to approach the schools. You might be surprised at what the offered school tells you. Even if you're not, you'll be more informed.