I'm so glad you aren't using a solicitor. There really isn't a need to. Try to remember that the panelists are all volunteers who have been trained to consider each case fairly and impartially. If, at the end of the appeal, they genuinely believe that adding another child to the school will cause more harm (prejudice) to the effective education of the existing children at the school than it will to refuse your child a place, you will lose, no matter how eloquently you argue your case. If, however, having heard the arguments and seeing any evidence, they genuinely believe that the harm (prejudice) to your child by being denied a place at this school outweighs any difficulties it may cause the school in fitting him in, you will win, no matter how faltering and inelegant your argument is. Because that is the job of the panel.
Remember that you are arguing for the appeal school. Try really hard not to mention portacabins, rough areas, high crime, new schools, etc. - The official stance is that all schools deemed operational are suitable to educate all children and a panel cannot be seen to agree that a school is unsuitable for a child, so don't try to convince them. Instead, focus on what your school of choice does do, does have, does provide, does offer, etc. - positive, positive, positive.
You can ask questions in writing and the school should provide the answers. If they can't, the LA must. Send a polite email.