When we asked for financial help we were told we'd get petrol money if we drove him which we can't do.
That is not good enough. Given the distance involved the LA is under an obligation to provide free transport if that is the nearest school with places available.
but would only help if they were paying for an official school bus (there is none) or X pence per mile if I drove him there which we can't do
So if parents can't drive and there is no school bus they think that somehow relieves them of their legal responsibility? Ridiculous. Tell them that you will be referring this to the LGO unless they refund your transport costs to date immediately and provide a bus pass for your son.
you have to ring each school individually yourself and ask if they have spaces
Also not good enough. Whilst the LA can insist you apply to schools rather than through the LA, they are still required to tell you which schools have places (Admissions Code paragraph 2.21).
Witchend is correct that normally your focus should be on the school you want. However, when dealing with an in-year application it can be worth bringing up bullying if you can show that your child has been bullied and the school has failed to handle it correctly. The point is to show the appeal panel why the child cannot remain at their current school. Having said that, you only mention one incident and, whilst they took their time, they appear to have dealt with it. So overall I agree that you should focus on the transport. However, if there have been several incidents and you can show evidence that they have not been dealt with properly (copies of emails, for example) that would be worth bringing up. If you can show that the current school is giving false evidence that would also help - even if you can't disprove everything they say, disproving some of it throws doubt on the rest.
Government guidance is that a journey of up to 1.25 hours each way is reasonable for a secondary school pupil. This journey clearly takes longer than that so your argument is that it is unreasonable, you need a school nearer home and the LA is not helping you find one. As the LA has acknowledged that this is the nearest school with places available the fact that you have not applied to other schools should not be a problem.
You can strengthen your appeal further by looking for things the catchment school offers which are not available from your son's current school that are relevant to him. For example, if he is musically talented and the catchment school has more extra-curricular musical activities that would help.