I find it helps if you try to see it from the LEAs/panels pov.
Their remit is to be equally fair to all children, which may mean giving priority to children with special social and medical needs to ensure a level playing field; it will also mean not admitting extra children unless there are very serious considerations as that would disadvantage the children already at the school. So all the time they are looking at the needs of the children, not the parents. And with a primary school appeal in particular, they have to be very careful about the bit about not disadvantaging the children already at the school: this is what the PAN thing is all about.
Special considerations can be things like a disabiled child needing a school with disabled access, a child in foster care needing to get into the school which is not in the same road as the house of his abusive mother, a child with a terminally ill parent needing to get into the one school that has a specialist bereavement counsellor. All these needs have to be proved by external evidence from professionals: it is not enough just to say that you have these needs.
The problem with your arguments is that they are all about your convenience- and tbh they would be difficult to prove. If they really wanted to get into one particular school, most parents could probably come up with something similar, something they just had to do at that particular time, and how would you decide between them? Many, many parents are in similar sitautions to you, in that they have to deal with another child (feeding, taking a sibling to a different school etc) at the time of the school run. We dealt with it by getting a childminder to take one dc, others deal with it by involving friends or organising a school run rota.
Anyway, it is a very temporary situation: your baby's feeding routine will change many times over the next year or so, so it's hardly enough for the panel to make a decision that will affect the whole schooling experience of the children involved.
From the panel's pov they are looking at how they can be as fair to little Johnnie and Kylie as possible, not to Johnnie's and Kylie's mum and dad.