I had the Sunshine Mimosa for a few years but took it out. It looked good in the summer but during the winter it was very sparse and the weeds would pop up between the runners. I put in some Alba Begonias in that area, time will tell if they do well.
My neighbor and I planted some Baby Sun Rose ( Aptenia Cordifolia) in our swale where it was hard for grass to grow. It's spreading slowly but we like it so far.
Perennial Peanut is another ground cover that you could use and you can mow it if necessary.
I would remove the grass but then I always do things the hard way. Someone else might have a better suggestion for you.

Perennial peanut looks ratty in the winter dry season if not watered sufficiently. Asiatic jasmine will work well in shade or sun, but doesn't like heavy foot traffic any more than perennial peanut....

Tom

Tricolor in shade:

Green, in sun:

I love my Dwarf Chenille ground cover. Soft on the feet and springs back when walked on. Those red fuzzies are adorable. I only have them in a year and they bloomed all winter, but it was a mild winter. I have no idea how long they will last. They do choke out the weeds. Occasionally I have to pull some. I simply cut out a spot with a scissor and plant something in between. They get along with my Bromeliads as you can see.

closer

even closer

Halfway, I love the look of that dwarf chenille in that bed. Is that in full sun or part shade? How difficult is that to keep trimmed back?

michael z9 its a bit of both. The tall columns and palms you cant see provide shade but a lot of it gets no shade here in Ft. Lauderdale area. It does well!