It's probably easier to appeal for a non-selective
school without the support of your primary school (because it isn't usually needed). You don't have
to be in catchment, but you do need reasons for wanting a place at the school:http://www.elevenplusexams.co.uk/appeal ... -school#c2
Inevitably, the more popular the school, the more you're competing with other parents for a place.
With regard to the selective schools, it's not impossible to appeal without support from the current school, although it probably makes it more difficult.
See B20(a):http://www.elevenplusexams.co.uk/appeal ... cation#b20
Appeals for any super-selective tend to be hard to win anyway. You may have seen this thread:viewtopic.php?t=19293
I suggest your next step is to talk to the current headteacher to try and pin the school down with regard to how much support they will give. The following link shows the ideal 'normal' range of evidence (school support included):http://www.elevenplusexams.co.uk/appeal ... cation#b11