Let me guess. The tree in the background is south of the problem area? Or put another way the problem grass is on the north side of tree?

If that statement is true, there is nothing you can do, the grass will die. As the sun gets lower in the southern sky the area will grow with the growing shadow. Your house completely shades it out half the day, then the tree blocks light when the sun clears the house.

Bermuda is full sun grass and does not tolerate shade.

The crepe myrtle tree is to the East. I have a friend whose yard is all shaded and her Bermuda grows but thin. To the right of that crepe myrtle is thin, but this area gets a lot of sun considering privacy fence and two houses.

Was the sod rolled down or trampled down? It looks like it was not in that one area. It is too late to roll it down now, but at least there is some bermuda. It should spread back in over the next few months. It seems to spread faster with just slightly cooler temps.

Where do you live?