I usually cut mine 1 notch lower than where i want my grass height. Kind of a small scalping in late July early Aug.

If you are at the bottom notch on your mower, then you could rent a mower that will mow lower and take off a little more stem. Or mow as low as yours will go once and raise it one notch to leave a little blade when you mow.

If you can mow low enough, that stem will grow out along the ground instead of up. That's when bermuda really look great.

The mower height isn't an issue. I've got quite a few more notches I can go down. The question is how to reduce the height of the stem, which is about an inch long with a single blade growing off of the side and three blades coming up out of the top. Even if I were to follow the 1/3 rule, I would eventually get to the point where I were cutting all of the top blades leaving only stem and the lower single blade.

If you cut down to the stems, you'll have brown stubble which will grow new grass blades at the lower height. It will be unsightly for a while which is why most of us do this in early spring. I was a little late this year as I did it in late April. The lawn has already recovered.

Mower height isn't an issue? No, but it is the solution. Unless you want to chew it down by mouth. To reduce the height of grass in a controlled fashion, the lawn mower is the best tool around.