Due to the trees, Zoysia may be a better bet than Bermuda. Why spend a lot, only to have it die when the trees start giving more shade. Besides the bermuda right under the trees will die anyway. Plant el-toro zoysia, which can take shade as well as sun, and produces less thatch. St Augustine requires more water, and more care as well.
Zoysia adapts to the limestone type soil pretty well.
You may need to have an inch of topsoil put before laying down sod. Due to the trees, Zoysia may be a better bet than Bermuda. Why spend a lot, only to have it die when the trees start giving more shade. Besides the bermuda right under the trees will die anyway. Plant el-toro zoysia, which can take shade as well as sun, and produces less thatch.