How to appeal a place for reception class at va school

[Markhampstead]

My eldest (going into year 2) is being offered a place at a local school but I'm struggling to find one for my youngest who is going into reception.
I have explained my circumstances on another thread but if you have not read it this is our situation.
My wife passed away last year and we are moving to hampstead over the next few weeks. Camden have agreed after providing a letter from our gp that they will be classed as special medical. I have sent a letter asking if they would allow my youngest to attend and they oversubscribe by 1, they have had the class sizes reduced to 27 for the whole school and the head has said it looks very unlikely and not to get my hopes up. There is another school just around the corner and it looks of similar size if not smaller but I can't be sure (hampstead parochial). I fear not getting them into the same school (just like every other parent) but I feel splitting them up or sending them to a large school will cause them a lot of upset as the school they attend now only has 200 and they are quite delicate at the moment and need the caring feel from a smaller school.
I know we are not the only family to suffer this way but would like to think even though I have been told it looks unlikely that if I put a good case across it might happen, am I being realistic or have I not got a chance?

If the classes have 27 pupils that is good as it means this will not be an infant class size case. Check that carefully.

On the assumption that it is not an infant class size case, you can win by showing how your child will be disadvantaged if not admitted to the appeal school. It must be about how this will affect them, so things like transport and childcare difficulties don't count. You need to show why this is the best school for your child.