"Exceptional circumstances" (or "Extenuating circumstances") is a phrase normally used in relation to underperformance in the 11+.
I'm going to assume that you're talking about an oversubscribed non-selective school, so please correct me if I'm wrong.
Some - not
all - admission authorities have an "Exceptional Social/Medical Criterion".
Have you checked the admission rules for the school you want?
Even if they have an "Exceptional Social/Medical Criterion", it is likely to be enforced very
rigorously. There would need to be supporting evidence that this is the only
school that could possibly meet your son's needs.
I'm sorry to say I think your LA are just palming you off by referring you to the "reasons box" on the CAF form. The box is only there because the government says it must be! In reality it forms no part of the admissions process. If there is a sibling rule or an "Exceptional Social/Medical Criterion", there are usually specific boxes to be ticked, or an instruction about what to do.http://www.elevenplusexams.co.uk/appeals/general#a27
Places can only be allocated in line with the published criteria. Those published criteria are not going to be changed because of anything someone writes in the "reasons box".
However, at a later date, an appeal panel would be interested in any reasons you originally mentioned.
Without supporting evidence, though, it might not be a strong case.
The deadline for submitting evidence for appeal purposes is usually quite late - probably sometime in the early summer term (you should be given an exact date in due course).
If the redundancy goes ahead, and if there is a serious impact on your son, one would expect the current class teacher to notice that all is not well. A strong letter of support from the current school might well help.