Both sunshine mimosa and frogfruit are good options for a low-maintenance ground cover in your area since they're both tough and can handle the heat and full sun well once established!
I think its flowers are a huge bonus, but sunshine mimosa can be a little slower to establish at first.
You might need to do some weeding around it until it really takes over.
Frogfuit on the other hand is great at outcompeting other stuff, and it stays pretty green year-round!
It'll spread fast, so you'll want to keep an eye on the edges, but that's a small price to pay for not mowing. It also handles foot traffic better than you'd think.
Either way, you'll be doing a lot less work than with that zoysia.
Just keep them watered well in the beginning and they'll take care of the rest!