I dealt with this on my Series III that I used to play exactly as you describe it. G# was the only note that broke octaves for me. I opened up the body octave vent hole a touch (drilled it with a Dremel) - didn't notice any difference whatsoever. What I really needed to do was use a more open tip mouthpiece, stop pinching, and open my throat. I do horn refurbs and don't recommend that most people take their horn apart and take a drill to it, but I was willing and comfortable enough to tackle the task. I play Keilwerths now and I had to adjust due to this on E2 breaking to E1 if my embouchre wasn't quite right. But with practice, I sorted out it was more me than the horn and I've overcome this problem without modifying the horn. And my embouchre has improved through the process of sorting this all out, so maybe give that a try. Also, it doesn't hurt to clean out the body octave vent with a bent paper clip or pipe cleaner (down through the neck receiver and through the body octave pip). Good Luck