There isn't much value in doing so.

The USA doesn't have any disagreements with other countries in the Caribbean regarding claims to the sea like China has with its neighbors.

China would need to invest a significant amount of funding into arming Cuba in such a way as to enforce an embargo of the Gulf Coast. Even then, it isn't anticipated that Cuba could hold it long to an invasion. If Cuba could hold on, the USA could still use its other two coasts to ship goods.

And it isn't like China doesn't have economic access to the region.