I understand that you possibly didn't factor in the logistics when you chose this school (and you did choose it, by putting it 3rd) but you seem to be stretching the truth to try, and find some wiggle room, to change your school choice.
I strongly feel your using DF cancer as an appeal, when previously you have said DM for childcare, is unethical. He didn't factor in your original plans for getting DS to school but now you feel his cancer will hold more weight at appeal, suddenly he's involved.
People have given you excellent advice, I am often amazed the time/effort/energy they put in assisting people through all this stress but you haven't listened, only tried to find a loop hole in what the are advising.
The fact is GPs do school runs, people get sick, have babies, have religious needs, have musical children. None of these are unique, many people have far more severe situations and they are still not grounds for appeal.
I'm a single parent with no family, 3 young children, one with an acquired brain injury and wouldn't use any of it as grounds for appeal. A mum at school was undergoing Chemotherapy when her DD started in September, she had to walk as couldn't drive because of chemo and didnt appeal her other, closer, schools.
Life is difficult some times, we don't get what we want and have to deal with the consequences of actions we didn't make. I really feel, as a Christian, we have to deal with things in a morally ethical way and in this situation I feel that is to find the solutions and not fight the decision.