Taking what other posters have (it seems) unearthed from other threads at face value and with one or two points from earlier posts here, this is what I’d suggest ...
The admissions process works on the assumption that schools can cater for children with a diverse range of needs. OP will therefore need to persuade the panel that her child’s needs are such that the allocated school can’t meet them but the preferred school can. She therefore needs to highlight those aspects of the school that make it uniquely suitable.
On top of that, assuming this is an infant class size appeal (which is winnable only if there has been an error depriving the child of a place or the decision to refuse is unreasonable in the legal sense), she will need to persuade the panel that the refusal of a place was so unreasonable that no rational LEA would have made the same decision. That rarely happens, but OP might manage to do it, especially if she can demonstrate that the LEA overlooked evidence from medical professionals setting out why the preferred school was uniquely suitable.
Talk about being “aggressive” is misplaced. It’s about presenting clear, factual information. It’s probably also misleading to talk about winning the appeal “the first time”. Generally, there is only one time, although there are some limited circumstances in which cases can be referred to the LGO or equivalent for academies. The right of appeal can be used once a year, so if the first appeal doesn’t succeed there’s a long wait until the next appeal for the same school can be made. In the meantime, if the child is of compulsory school age, they need to be receiving an education in some way; in this instance, as OP has said that private school and HE aren’t options, that implies looking for another acceptable school. That may not be ideal, but I can’t see any other option.