There are very limited grounds for successful appeal under ICS rules
Essentially
A) the entrance criteria were in some way unlawful - not giving priority to a group they must (LAC) or giving priority to a group they must not. This is very rare. It does not mean that 'they could have done it differently within the code' - admissions authorities do not have use every possibly criterion, but once criteria are published for that year they must stick to them. Blatant gerrymandering of catchment areas (defined priority admission areas) could count as this.
B) there has been a mistake in processing the application and that error has cost your DC a place they would have been offered if no error had mpbeen made
Have you checked that they placed your DC is the correct oversubscription category and that they have the distance (usual tiebreaker) correct?
C) the decision is so unreasonable that it's perverse in a legal sort of sense (no rational person could possibly have made it) - the threadhold for this is very high, child protection issues or significant mobility issues (like placing a child who needs to use a wheelchair in the only school for miles around which is not adequately accessible)
Your case looks like a long shot, but it is still worth trying. Check if the school had an exceptional social need category - some LAs will place major mobility issues in the primary care giver into that category. It won't win you an appeal, as the council did not have that information when they processed your application so it's not an error, but you can get it lodged with them (outside the appeals process as well) now and maybe move up the waiting list if it moves you to a higher category
Wraparound care, or other school run logistics, really won't come in to it, sorry