Try to think of it from the panel's pov. They most likely know that one school is not as good as the others. They can't let all the children in town into the good school, because then it wouldn't be a good school any longer.
So apart from mistakes in application procedure, the only other way they can let one child in, is if you can prove that there is some special reason why this child deserves or needs a place more than other children. And why the prejudice to your child from not getting this school would be greater than the prejudice caused the children already accepted by cramming more children into a school that is full.
The problem with the better school argument is that the LEA won't have any reason to think that your child deserves the better school more than any other child that they won't be able to admit.
Social reasons I have known to work are:
one school offering special counselling to a bereaved child or a child about to be bereaved
an abused child needing to get away from a certain area because of the people who live there
a child who has suffered trauma needing to go to same school as friends who can support her
But there is always the special aptitude argument too. Be aware that you'll need to show some proof of the special aptitude though.