Im no expert, but ive heard shading it out is the best way. One guy laid black plastic over the area and put about 6" of mulch on top. It should keep it from getting any light so in return it kills the grass. then just rake up the area and re sod or wait for your st augustine to grow back in. Im no expert though but it makes sense to me. No light = No grass.

If you are going to let drought come and go without watering, then bermuda is the dominant species. If you irrigate weekly and mow at your mower's highest setting, then St Augustine is the dominant species. By mowing high you are shading the bermuda out without doing anything extraordinary.