What part of a Cattleya orchid is this?

That's a seed pod- the result of pollination of the flower. On the far end with the withered sepals and petals, you can see the still-green column which was enclosed in the lip while the flower was still fresh.

 

That is a seed pod (seed capsule). When mature, it can contain hundreds of thousands of dust-like miniscule seeds. To germinate, orchid seeds require the presence of very specific mycorrhizae. Even under the best of conditions only a few will grow up to become orchid plants. The odds of getting sprouts even if it had been left to maturity, would be very long, indeed. However, living in a tropical climate as you do, the next time you get one you could leave it to develop, you might even be surprised. There's certainly no harm in letting it "go natural".