My question is this: with the approval from my flying school, is there any reason why I can't go to another flying school where there is an increased likelihood that the QXC can go ahead? I'm thinking mainly in the South (East) of England where geographical relief doesn't affect the weather as much as where I'm based.
Legally you would not even need approval from your flying school. The only legal requirement is to have a number of training hours when you apply for the PPL, but it doesn't matter where you got those hours from (provided that all the paperwork is in order and all hours are accounted for).
But practically speaking you not only have to arrange informal approval from your own school (and you need to be very careful in how you phrase/obtain that - this is the sort of thing that can break a relation very easily) but you also will need to fly a few hours with that other school before they will be confident enough to send you x-country solo. After all, you are now the responsibility of that other school.
That's not even considering different planes, different airspace, booking procedures, faces and so forth.
My advice would be not to try this if it's just for a cross-country. Instead, talk to your employer and arrange that you can have a day off at (very) short notice. Keep track of the weather, if necessary with help of an experienced pilot or instructor. As soon as the weather looks right, book a plane, get your instructors approval and go fly.