Assuming your son is in Year 3 or above, this should be a normal prejudice appeal (assuming you are in England or Wales).
This means you have to persuade the panel that the prejudice to your son of not attending is greater than the prejudice to the school of taking another child.
Contrary to what PP have said, this doesn't have to be anything specific around social/emotional/discrimination; it can be anything that shows the prejudice. In reality, most successful appeals offer a combination of reasons, unless there is something hugely standout, which it doesn't seem there is here.
The fact the class had one over and a child has left doesn't mean there is a space just to hand out; it sounds like the class was one over PAN, for whatever reason (a previous won appeal, for example, or a child admitted late with an EHCP naming the school) and now a child has left which has taken them back to PAN. They don't have a "spare" space until they go back under PAN.
However, the fact (possibly) that the class has operated fine with 1 over PAN is useful, as it demonstrates that they can cope.
The head should not support the appeal though, so don't expect a letter. He may not object too much, but assuming it's not an academy, it's not up to him.
You need to evidence that the school can cope with one more, and that the detriment to your son of not being admitted to this school is greater than the detriment to the school of having to cope.