Let's be honest, the loveliness of the fence is a matter of taste. Personally I think it looks a bit twee. The comparisons with the chocolate box cottages don't work for me either. Twee little picket fences belong on twee little chocolate box cottages, I'm not sure they belong on estate council houses - well, not with finial topped posts anyway.
It's been done to death now but I agree that you shouldn't have put in a planning app. All you've done by doing that is give them something to work with. If you hadn't they would now have to think about enforcement which would be almost impossible to justify. Still, done now so what to do?
Personally I'd plant some shrubs in front of the fence. You've got a bit of space there to do that and the right ones will soften it up and take the harshness out. I'd also cut off the finials. Then appeal but don't employ anyone - just do it yourself. You can do it online and there's no fee. Whatever you do don't lay it on with the attack and how the neighbour's a nutter etc. Just say the application was refused and you disagree with the reasons. If you can find any similar local fences it wouldn't do any harm to submit a photo with the location but don't go mad. With some planting and the finials gone I'd be surprised if your appeal isn't allowed.
PS. If you do appeal, before the inspector visits, you might also consider toning the whiteness down a bit. Spray it with slightly muddy water. You an always clean it up afterwards if you prefer the unnaturally bright look.