As always the big question is whether this is an infant class size appeal (assuming you are appealing for a place in Reception, Y1 or Y2).
Regardless of whether or not it is an ICS case your own anxiety issues and panic attacks are not relevant I'm afraid.
If it is not an infant class size case the bullying, etc. may persuade a panel especially if you can provide evidence to show your family is being targeted by people with children at this school. Even if it is an infant class size case an appeal panel may take the view that it is unreasonable to refuse to admit your child. So yes, you have some good points.
Without knowing exactly what the letter from the teacher states it is difficult to judge how much that will help you. In general friendship issues don't carry much weight in appeals unless there is expert evidence to show that the child has significantly more than normal need to stay with existing friends.
I would agree with the advice you have been given that you should also talk about what the preferred school can offer your son that is missing from the offered school.
Don't worry about the terminology. It isn't really important. If you need anything explaining feel free to ask.
"Exceptional circumstances" are situations that are very much out of the ordinary. So, for example, the child of a social worker being allocated to the same school as the children with whom their parent works may be regarded as an exceptional circumstance that justifies moving the child to a different school.