I was like that for years. The problem is learning to manage your regime while doing sport. I recommend Diabetic Athlete's Handbook by Sheri Colberg. It's not necessary to go LCHF once you get the relation between insulin and exercise sorted. My Hb was like yours because of the struggle with hypos. I ended up keeping it high as a survival strategy, but once I learned how sport and insulin interact (no doctor or DSN ever helped me with that), basically all my problems were solved. My Hb is now in the low 6s, and I'm not constantly struggling. One of the issues is that you are probably very insulin sensitive because of the sport, but doctors keep telling you to increase your insulin because of the Hb. But that doesn't get to the root of the problem. Since I started working it out with Colberg's book, as well as my Hb going down, my total daily dose of insulin has gone down from 40ish units to 24-26 units per day.
If you are in the UK, things are better these day, though. I believe the DAFNE course and other such courses can help you.