You don't need to prove a mistake.
You just need to prove that your child will suffer more detriment by not attending that particular school than the other children will by now having a bigger class, less equipment to go round etc.
Its hard. Not impossible but hard.
You cant appeal against a school so talking about the bad experience etc won't help. You can only appeal for a school.
What does this particular school offer that your child needs that can't be provided in anyway shape or form by the allocated school?
With regards to the other school-
Accept the place. The LA has done its job. It is under no obligation to find u another one. Declining it will leave you without a school or you will end up somewhere equally unsuitable but miles away.
Do you have the full story for the bad experience?
ImE all the parents thrilled about getting a particular school as it was highly sought after do not appear to have fewer complaints than those who got the less desirable schools. In fact some were extremely shocked at how things realky were. Many of the good schools are coasting on reputations from years ago. The reality is still lots of supply teachers ,.poor SN provision and buildings that are in sorry states etc that is everywhere I'm.afraid.
What other local schools do you have. Go on their waiting lists but check how many the LA will let you go on.
And visit the school you got. See for yourself.