If I were you, I would visit a botanical garden for ideas and see what the gardening professionals in your area are doing. Mounts Botanical Garden of Palm Beach would be a great resource. It may be that you want to garden up closer to the trunk of the tree and have open lawn where the branches fall. In the garden area by the palms, you could plant ferns, bromeliads, grasses, smaller palms, begonias and perennials. I hope you try a Pink Frost florida Anise-tree under the shade of your palms. Your local garden center would also be a good place to visit and get recommendations.
Watch out for falling branches!