Carefully cut them down and immediately paint the stump with RoundUp. It will resprout, but either brake the sprouts off or spray them with RoundUp. Suckers will also come up from the roots, but keep mowing them. Eventually, without leaves for photosynthesis, all will die. But it is an ongoing battle for a while.I usually never use Roundup, but do for locust and Johnson grass & poison ivy. (I am allergic to locust thorns and poison ivy, and you never get all of the roots of Johnson grass the first time.)

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Everything I have read about the Lady Banks seems perfect for here as far as the heat,etc.; but what about the heavy red clay? Honey Locust or even Purple Robe, are my favorite two locust, that I have grown in other climates. They will grow in harsh conditions, they are brittle, subject to wind damage, but I love them. I just want to be able to plant some things that will survive in the native heavy clay soil, without the trial and error. I have moved and planted my sweet woodruff by my front door at my last three homes, that's one thing that will give me a sense of home, in a new yard.

I see a lot of these trees around this area and for sale at the greenhouses. 2018-2019 was a brutal winter, if I remember correctly. It was after that winter that some of the smaller dead branches showed up, I don't remember it not completely leafing out last summer. Maybe it was like this and I didn't notice; I was really busy last summer and didn't spend a fraction of the time in the garden that I normally would.