FWIW, my weight has gone up and my fitness level has plummeted over the last six months as my job went from demanding to impossible, but my SAC rate stayed the same. Now that I've quit that job, started eating better and exercising, and stopped drinking so much, my blood pressure has returned to normal, I've started losing weight, I'm sleeping better-- and my SAC rate still hasn't changed. Maybe with really dramatic weight loss it would. But I think the key there is finning technique, streamlining, proper weighting, and comfort in the water. My SAC rate all this past year has been much better than it was the first year I started diving.
I also agree the key to weight loss is diet, not exercise. Exercise is worthwhile for many reasons, but as they say, you can't outrun a bad diet. Some people do well with calorie-counting apps. Others do better with diets that eliminate high-calorie foods so they can just eat as much as they want from a limited list. Still others are having better luck with intermittent fasting. The important thing is to find something you can do long-term, because once you stop, the weight tends to come back.