The class size thing is called 'future prejudice'. Appeals panels aren't allowed to direct schools about how they organise their classes. A parent may say "Well yes, there is a problem if you have mixed year groups, but I know you have a spare classroom and so you could employ another teacher and have single year classes." The panel can't say that it is a great idea and the school must do that.
The information that the panel has to consider is the PAN (Published admissions number) because the school is bound to admit up to that number. In fact, if someone appealed and the panel found that the number admitted was under PAN, the appeal would be successful straight away.
For this same reason, schools can't admit over PAN in one year to compensate for being under PAN in another because if a pupil from the under PAN year applied for a place they would still be bound to admit that pupil.
The question of numbers is not 'do we currently exceed Infant Class Size regulations?' It is 'if the school admitted up to the PAN, would admission of a further child breach the ICS regulations?' For your child, if they admit 15 per year group, if they used mix year classes, they end up with a year 1/2 class of 30 and that is the problem.
With regards the prohibited steps order, I can't see how being at a closer school protects your DD. The school policies and procedures are what protects her. Unless the school contacts you, you would be none the wiser at either school. Once the school procedures kick in, it wouldn't matter how long it takes you to get there, your DD would be safe.
Taxi transport is the norm for many, many children. For example, 90% of special school pupils arrive at school by taxi. In rural areas, most children get the bus to school from reception year. Because the allocated school is over 2 miles away, your DD will be entitled to transport. This is where your medical evidence will help determine the best mode of transport.
A panel is very unlikely to find that it is unreasonable to not grant a place on the basis of your arrangement with a neighbour. If you have no family ties to your current location, could you move closer to the allocated school? That would seem sensible.