The admissions appeal code says that schools should not weigh in on behalf of children in the appeal process -
1.29 ... Where the admission authority is the local authority, one additional representative of the school (e.g. a head teacher or governor) may accompany the presenting officer to assist proceedings, for example, by answering questions raised by appellants about the school. However, that additional person?s role at the hearing is solely to answer questions put to them about the school; the panel chair must not allow them to comment on the child, the appellant or their case.
So, even if an appellant has a letter of support from the HT or the governing body, the panel should not give it any weight.
Bothcholma - Is yours a school with a catchment area? Do you live in or outside it? You need to double-check how your twins came not to be offered places - presumably because there were so many other siblings living closer to the school or within catchment if you live outside it and fall into a lower priority group.
DAR82 - You mention that your child is baptised but you don't say whether you are a church member. Many C of E schools have admissions criteria which prioritise regular church attendance - is this where your application fell down? Similarly, some C of E schools reserve a proportion of their places for the community, so that might explain why a Catholic child was admitted before yours, especially if they live closer.
In any of these scenarios, the scope for winning an appeal is very limited if infant class size rules apply (generally, where there are 30 children in a class).
Refusing the place you have been offered does not give you any extra leverage at appeal - panels will not respond to blackmail.