Do you live in London Bexley or London Greenwich?
You are clearly bang on the border. You need to telephone your council. By all means call the other council as well to find out which schools in their borough have spaces. And you better add Bromley to the mix as well. Montbelle is Bromley I think.
Then I believe you need to go back via your council and tell them you wish to apply for those schools with places.
I cannot believe that the whole of the London Boroughs are full up from Bexley, through Bromley, Greenwich, Lewisham, Southwark and onwards. But it could be possible.
So guessing it isn't possible, and the council have made a mistake, the only mistake they have made is not giving you a school place closer than Kingston.
You will not get a place at Dulverton on appeal. Bexley council have not had any successful appeals in many years because they admit correctly on distance. Unless the mistake is about how far away from Dulverton you actually live. I expect the catchment will be somewhere between 0.5 -1.0 mile BY PUBLIC HIGHWAY. It is not as crow flies. For example if you are on the wrong side of the railway line.
I think the issue here is as you now know, naming one school on the form. You should have been given a place at the next nearest school with places after all other children have been given their place. The computer is saying Kingston.
You will not have to travel to Kingston. That is ridiculous. If all the locals are full, there would be a bulge class somewhere closer or you would get a place as an excepted 31st child. Can't think what this is called.
So there has been a mistake. But that mistake is not going to get you a place at Dulverton. Or Wyeborn, Longlands, etc etc.
There will be spaces somewhere closer in Bromley/Bexley/Greenwich. You will be given one of those.