It is difficult for a LA to argue that the presence of one more child will prejudice the efficient education of the others. It does become more difficult if every class of 10 in a special school already has 12 or 13 children in it. Then the latest child is becoming the 3rd or 4th over the numbers. However, it is open to the LA to put extra staff in.
The DfE website has the official numbers of children for each school on it (school capacity). So long as the number of children in the school is below the official capacity, then the LA cannot argue the school is full, even if the year group is full - subject to the question of 12 or 13 in a class as above.
I'd put the ball in the LA's court - point out the statement specifies a special school and they have a duty to implement the statement as it is, until they have, if they desire done a reassessment; and they could if necessary put extra staff in. I'd ring each special school nearby and ask how many children they actually have in total, as against how many they are registered for.
I agree, I'd also start asking the LA about specialist schools - which can concentrate minds.
I would also speak to Maxwell Gillott, or one of the other organisations with a franchise for legal aid in education, as the child has the right to seek judicial review if they are not getting the provisions of their statement - and most children, unless independently wealthy qualify for legal aid. Its usually quicker than the LGO route.