Has the new LEA offered you no school at all, or just not the one you want? If you fulfil the criteria for a school and there are places available in the relevant year groups, I'm not sure that they can refuse your application, but you need to look carefully at the LEA's 'rules'. Could there be an issue with the proximity of the current school?
If you are both at work all day, how would you actually home educate? I know that there are people who argue that e.g.the LEA shouldn't be allowed to 'interfere' in what one is doing in the name of home education, but you have a duty to provide some kind of actual education rather than just deregistering them from school. As for the 11+, though, this shouldn't make any difference, as there are bound to be other DC taking it who are home educated.
Outside of a dog, a book is a man's best friend. Inside of a dog it's too dark to read.Groucho Marx