Choose the grass you want. Mow high, +3.5" to shade out the bermuda and the SA should increase over time. Mow low 0.5" and the bermuda will takeover. You can add some chemical helpers if like that will harm/kill the SA faster but other wise will take a few growing seasons whichever grass you choose.

Exactly right. You can encourage St Aug to take over by mowing at the highest setting and skipping a week or two now and then. Water deeply every week in the summer. It will spread best when the temps are in the 70s and low 80s. It should creep 10 to 15 feet per year. Here are my most recent example pictures.

This picture shows almost no St Augustine. Most of the turf is bermuda, prairie, crabgrass, and some horse herb. If you look carefully the St Aug is between the pot and the concrete. There is also some underneath the faucet. That was summer of 2014. This is the same area with the pic taken last week. Everything that is lime green is St Aug (no fertilizer yet this year). The gray-green is prairie grasses. Knowing that I wanted to keep the St Aug I only watered the small areas where it was and let the other grasses suffer. When I saw a St Aug stolon creeping out into the other grasses, I watered that runner.

If you have common bermuda, I would let the St Aug take over. If you have hybrid bermuda, then you have a decision to make. If you want to keep the bermuda, mow it low 2x per week and stop watering. After a month with no water, the St Aug will begin to die out. As long as it has a golden cast to it, it is still viable, so be sure it is dead before you resume watering. The dormant bermuda will come back fine and fill in where the St Aug was.