If you have full sun, IMO Bermuda grss will win. You can try to cut you grass high (3 inches or more)to favor SA and weaken the Bermuda, but your chances of ever getting rid of Bermuda are slim to none.

Mu suggestion is if you have full sun, mow low around 1-inch, fertlize regularly, and enjoy a Bermuda lawn.

It is all about proper maintenance. Both grass require different maintenance.

For st augustine

1)Mow high - 3inches min. I mow mine at 4 inches once a week
2) Water deeply when needed. My rule of thumb is water deeply (1-1.5 inches) only if the grass start to wilt.

Bermuda grass

1)mow low and often. Usually 1 inch for rotary mower. 2-3 times a week.
2) same as above to look good.
3) More fertilizing than st augustine
4) Must be in full sun to thrive - will do poorly in shaded area where st augustine will thrive.

You just have to decide which grass you want to dominate. If you have trees, go with st augustine.

My soil and light are different from yours but I used to play games with the bermuda and St Aug that invaded my stone driveway. One year I would not water and watch the bermuda take over. Then next year I would water once a week and mow it high and watch the St Aug take over. I did that for about 10 years playing them back and forth. Now I'm doing that with some zoysia and bermuda.

Infrequent watering and mowing high has kept all the grassy weeds out of the sunniest parts of my lawn for years. I have had the occasional bermuda, nutgrass, and wildflowers (that thing everyone calls wild onion) but they have all gone away. Weeds I cannot control with water and mowing are broadleafed weeds like henbit, oxalis and chickweed.

Where do you live?