I think a first run across the Bermuda, with a shallow setting, and simply pick it all up (dirt and grass) and put it in the compost, then go back and till what is underneath every few days for a week or two might be one approach. You'd have to rake or pick up all of the roots you see each time, and then add in the compost and soil ammendments to build it back up again.

Where I lived in Seattle the garden we had when I was a child always started with the grass on top being turned under and that was that.

SRS