Start by trying to see it from the LEAs point of view (always helps to see the pov of the person you are trying to sway).
They have now allocated places, hopefully by following their own criteria.
They cannot accommodate your ds by throwing out another child who has already been offered a place.
So if they are to offer your ds a place, they will either have to stretch class numbers (not allowed above a certain number in infants) or employ another teacher. Which is going to come expensive.
You will therefore need to prove why your ds's need for this is so strong as to outweight the disadvantages caused to the other children/the school by doing so.
If you can prove that the LEA has failed to follow their own criteria, then they will have to offer you a place.
If not, you will have to prove that your ds has special social or medical needs which cannot be met by any other school. The concern about children being disruptive is going to be the same for all parents: they can't offer the whole county a place at your preferred school. Lots of parents are going to think their children are extra sensitive.
It needs to be something that sets your ds apart, and it needs to be supported by evidence, e.g. a letter from a paediatrician, educational psychologist etc. A parent saying that "my child is sensitive" won't be enough, lots of parents would say that to get their child into their preferred school. You need to provide evidence that your ds's sensitivity is such that he has been getting ongoing medical support. Otherwise, they'll just shrug and think "well, other sensitive children have to cope".
Our dd got into secondary (where they are allowed to stretch classes) but only after we had provided medical evidence from three different sources that she was wheelchair bound and that the schools offered did not have disabled access: we got turned down on our initial application because we only sent in a letter from her GP.