What do you want to know? No bribes needed! 
Appeal panels vary widely in the questions they ask. It is very difficult to predict. There are certain things they shouldn't ask, though. For example, they shouldn't ask why you didn't apply to school X or why you made school Y your second choice. They should concentrate on testing the strength of your case. Given what you have said I would expect some questions about your son's aspergers and how it affects him in daily life. The admission authority's representative may also ask you some questions.
I would have to say that you are most likely to be successful if you can show that a mistake has been made and your son should have been admitted. That will only fail if lots of mistakes have been made and the school couldn't cope with all of them.
Assuming no mistake has been made, a case with independent evidence from an expert showing that your child will suffer if he doesn't go to the GS is likely to be pretty strong. The expert's report needs to say "I have examined curlymama's son and in my opinion..." or similar, NOT "curlymama tells me...". The latter tells the panel that the expert is simply repeating what you have told him, so they will simply discount the expert's evidence. You have to be aware, however, that even if you have a very strong case, sometimes the case to refuse admission is so overwhelming that you won't succeed.
It is always difficult to predict the result of an appeal where the panel has to balance the prejudice to the child of not being admitted against the prejudice to the school of having another pupil. Different panels will arrive at different conclusions given the same facts. Sometimes a panel will admit a child even when the rules say they shouldn't. And sometimes a panel will refuse to admit a child even when the case to admit seems to be overwhelming.
Good luck and feel free to ask anything you want about admissions and/or appeals.