St Augustine will choke out bermuda easily IF you treat the St Augustine properly. It needs water or else it will die. If you are not the kind of person who will water during a drought, then get used to bermuda (which can be the nicest lawn on the block if you want to take care of it). Also to choke the bermuda out you need to reset your mower to the highest setting. Bermuda cannot compete with well watered, tall St Augustine.
I tried that and it did not work for me. The St. Augustine is fine in the shade, bet where there is more sun, bermuda reigns. My herb gardens are in the sun and in raised beds. Bermuda finds it's way in, regardless. Bermuda is evil, in my opinion.
I realize now what I must do. I need to take a sod cutter and cut it out along with 2 or 3 inches of the underlying soil. Hopefully, I have not already pushed rhizomes deeper into the ground while working the soil. If I have, the monster will surely rear it's ugly head at some point.