If you are watering deeply and infrequently, and mowing at the mower's highest setting once every week or two, the St Aug should be extremely dense and definitely dominate all the grasses you mentioned. I've converted a bermuda or weed yard to St Augustine several times. It's really easy. Here's a picture of my current project.

All the green in that picture is new St Augustine. Despite the optical illusion in the picture, it only covers about 1/3 of the yard. When I moved in 2 years ago the only St Aug was about an inch of it growing under the spigot in the shady corner by the brick. The rest was weeds and bermuda. Now it's almost too dense to walk in. This is in the Texas Hill Country so we don't get a lot of rain. I've watered it about 10 times and it's rained maybe 10 times in the 2 years. You can't see this years runners in this picture, but you will next year. They extend 10 feet out beyond the real green looking area.

Zoysia, on the other hand, is often more of a pain than any other grasses. The main problem with it is the inability to recover from damage or disease within a reasonable time. In my experience with it, once it gets stressed, it is down for the entire rest of the growing season and only comes back the following April. There are many other varieties of zoysia, so if you are going to do that, be sure you get one adapted to your neighborhood. But as a warning, the zoysia installs I've seen almost all got switched out for St Augustine within a few months. Those that did not get changed out were either because they sunk too much money into it or because they had unlimited resources to spend on fixing it.

Here's a picture of a house with unlimited resources to spend on their zoysia lawn.

The picture was taken in July 2013. The brown spot was caused by too much rain that month. A disease process got established and killed out the grass. The owner did not patch it and the grass returned almost normal the following year. But if you walk by that lawn today, you can still see, after 3 years, where this dead patch was. It's just thinner there.

If you have wild Bermuda, an improved Bermuda is the best option in my opinion. If you can't beat em, join em.