There does appear to have been some degree of exchange between the Antilles and the mainland, though I can't vouch for Mexico being part of it.

Professor Antonio Curet of the Museum of the American Indian was on a recent episode of the AskHistorians Podcast. He mentioned that we appear to find proof of trade through the fact that certain kinds of stone blades typical of Central American indigenous civilizations of the time were found in the Antilles while the sites where these blades are typically produced have not been found in the Antilles. Ergo, it is likely these blades were brought from the mainland.

The Taino do appear to have done some degree of trading and certainly had the nautical tech and know-how to do so.

It should also be noted that the Westernmost tip of Cuba was close to the Northeastern part of the Yucatan peninsula, still free from the Aztec Empire. Indigenous peoples there served as a buffer between the two.

The problem here is that we lack both written sources or historical memory to back this up. The Caribbean was populated by pre-literate societies whose historical memory was preserved by oral tradition. An oral tradition devastated by disease, massacres, and forced labor systems, like the encomienda. There has also not been near enough archeological excavations in the Western Caribbean.