Have the LA mentioned the magic words "infant class size" or "class size prejudice" in their case? From what you have posted it doesn't sound like they are making an infant class size case but it is important to be sure. If they have not mentioned either of those phrases this is an ordinary appeal which means you will have a better chance of winning. However, if this is an infant class size appeal I'm afraid you should only win if you can show that a mistake has been made which is unlikely.
Without seeing exactly what they've said, their case sounds fairly weak from what you have posted. There is plenty there about how they have followed the correct process but they don't seem to have said anything at all about the problems the school will face if your child is admitted. Normally I would at least expect them to talk about crowding in the corridors and communal areas or something like that. Here it seems all they have said is that they cannot continue to increase pupil numbers without justifying that statement. If the LA cannot show that admitting your child will cause problems for the school the appeal panel is required to decide in your favour.
Just to interpret their case for you:
admitted up to PAN
That basically means Y1 is full. If it wasn't they would have to admit your son.
catchment criterion no longer relevant
As an in year admission the only question they need to answer is whether or not the school has a place available. If it does and your son is the only child applying he must be offered the place. If there is no place available the fact you are in catchment simply ensures you get priority on the waiting list. It does not get you a place at the school. I'm not really sure why they feel the need to say this as part of their case.
PAN set having regard to net capacity assessment
So they have followed the law. The net capacity assessment is the number of pupils the school can accommodate if it is full up. The PAN for a primary school is usually the net capacity divided by 7 (as there are 7 years in the school) but they can set it higher or lower than this figure. Have they stated the net capacity of the school and the number of pupils currently on the roll?
places available at other schools in the area
If there were no places within a reasonable distance they would have to deal with you under their Fair Access Protocol, which may have resulted in you getting a place at this school. Again, not quite sure why they have felt the need to say this. The panel can see the allocated school so it is obvious that a place is available there. I'm afraid the fact it is over 2 miles away does not make the distance unreasonable. Note, however, that your son is entitled to free school transport if it is over 2 miles walking distance.
cannot continue to increase pupil numbers
Why? On its own that statement is useless. They need to justify it.
admission criteria objective
Again, that's just saying they have followed the law.
You can certainly talk about the damage to your son's social development if he goes to the allocated school but I wouldn't regard that as a particularly strong case. That won't matter if the panel decide that the LA haven't shown the school will suffer if your son is admitted but it is worth trying to improve your case. Look for any features of this school that are missing from the allocated school which would be of particular benefit to your son. For example, if your son has an aptitude for music and this school has a school band but the allocated school doesn't that would be worth raising.
If the LA's case does not say what problems the school will suffer if your son is admitted I wouldn't ask them any questions about that in the hearing as they may come up with a good answer. I would wait until you are asked to summarise your case at the end of the hearing and then point out that the LA has failed to produce any evidence to show that admitting your son will cause prejudice to the school.