Hello BusyIzzy. I'm sorry to hear of the difficult time that you and your daughter have had.
You don't say what the Published Admission Number is. Is it a multiple of 30? As Sunnyday says, if it is then the infant class size regulations will kick in, and (according to the admissions appeal code) you should only win your appeal if you can demonstrate that there has been an error or maladministration or the admissions criteria are so unreasonable that they should be set aside. If the PAN isn't 30 per class you have more scope for winning your appeal.
On the information here, I can't see any evidence of a mistake. As you probably know, attending the pre-school doesn't give priority for admission to the school and you are in a way paying the price for putting your daughter in a pre-school outside your catchment area.
However there are two things which might just help you. Schools are not supposed to hold back places for appeals, but the fact that the school has done so maybe means that the appeal panel takes a more lenient attitude towards allowing appeals than the law says they should. The other thing that surprises me is that there is no admissions category for social/medical need. You could argue that if the school had adopted such a category, your daughter would have fallen into it because the difficulties she experienced in her early years give her a strong claim for staying at the school for continuity and pastoral care, over and above any other child who's been in the pre-school. Can you provide letters of support from professionals (health visitor, police family liaison officer, court official, etc) to confirm that your daughter witnessed domestic violence and in their professional opinion she would benefit from staying at the school where she has made such good progress?
Don't worry that you and your daughter don't 'fit' the demographic of the school. That's not relevant to the appeal, which should be decided on its merits. Just go in there and do your best for your daughter.
One last thing. Winning an appeal does not mean that another child will lose their place. There are very limited circumstances in which school places can be withdrawn and another child winning their appeal isn't one of them. Children admitted on appeal are 'excepted' which means that they are admitted over and above the usual class size limit of 30.
Good luck.