Thanks to the vagaries of DNS the changeover took a while longer than I thought. In the end I had to export the database, import the database, copy the theme/uploads over and make some changes directly to the database to get things working. It seems a site of this size (700MB+) isn’t catered for by most of the cloning or import/export plugins available for WordPress.
Still, I’ve learned some valuable lessons about how my host operates and the importance of encoding in SQL import files so it was a valuable exercise in the end. Now I can start moving my client sites over knowing I’ve encountered most of the issues I’m likely to.