There are some real experts on appeals on MN, so I hope they will be along soon to help you. It does sound as if you're having a horrid time of it.
To help them advise you, could you clarify that this was an in- year application when you arrived (some time in October)? When were you informed that your preferred choices were full, when were you offered the place at the distant school, and when was the transport offer made?
Sending children alone in a taxi, no matter how much you dislike it, is a normal way of providing school transport. In offering you a school place and transport to it, the LA has fulfilled its obligations. If you declinedthe place, it becomes your duty to ensure your DS receives an education (eg HE or other private arrangement).
Are you on the wait list for all the schools you would find acceptable, and do you know where you are on those lists?
On what grounds did you appeal? I'm rather assuming that you do not believe they made an error when they said the schools were full. Is that the case? In which case, you have to show that the prejudice to your DS by refusing admission is greater than the prejudice to the school in admitting him. This is where the experts, and the outline of your case might be useful.