Well like we said, be careful with the Ofsted ratings.
I know you're only thinking primary school at the minute, but the 'outstanding' secondary school near us is rife with problems. If doesn't guarantee 100% no problems. The 'good' school we went to visit for DDs SS next year admitted to us personally that they too have their own problems. When you say undesirable, do you mean just public opinion or what is their Ofsted rating. If poor, this can sometimes be a good thing as it means they get extra help and manage to turn themselves around quite quickly with the right input from outside teachers etc.
Like I said. Mine was simply because DSs current school couldn't / wouldn't fulfill his needs, and my DD was being bullied horrifically (threafs of violence and being scared out of her wits when they tell her they're going to lock her in a dungeon overnight on a camp trip and no one will ever find her - these were the nicer incidents!). Not meaning to be trite, just straight, but if you do appeal you'll need a lot more reason than the nearest school simply being undesirable. I am still convinced the only reason we won is because the counsellor weighed in as to how it was affecting her and I also had a bloody good education solicitor.
Also try and think practically, do you drive? I don't and often have to get a taxi to the school and back when walking isn't possible (I have a smashed kneecap). The journey is just under two miles so we wouldn't qualify for travel costs either. But the help and support both my children now get is more than worth it for the hell they went through.
In reference to the home schooling, I had a friend who did just this and the council are quite hot on checking up you are teaching them right now I was led to believe. (she was basically told she wasn't doing enough and had to put them back in education- I'm not sure how common this is though).
Let us know how you get on 