The problem doesn't exist when you play loud (or at least not when I play loud).I worked on a Cannonball tenor one time that when the upper octave G was played very loudly, the low G also sounded producing a multiphonic effect. The advice I got from Tevis at Cannonball was to open just the bottom of the body octave pip a little with a small rat tail file. It proved to be a good solution.
Stephen, can you describe what you mean by the G "breaking up"?
To get the note to break (and turn into a harmonic squeal) I had to blow very softly.
I can't get my M2 Studio (same as Rhys's horn but with a standard-width bell) to do it.