I spoke today about this piece with an anthropologist-archaeologist who works at the Royal Ontario Museum, and she says that the item on the set Desilu set was not a copy of of an Aztec design at all. It is, rather, something far more obscure: a 1:1 scale copy of an ancient Zapotec urn, painted red. The Royal Ontario museum has a number of such urns, though not this one in particular.
It seems that as a result of our site and this article, one of the leading experts on the subject of Zapotec urns, Dr. Adam Sellen, is working on putting together some research and an article on the appearance of the urn in Trek. This is very exciting to me, and I hope he's able to track down what the original was! If it were possible to obtain photos of the original object of which the Kirk sculpture is a copy, it would make creating a replica much easier! And, in any case, I think it would make for interesting reading. If he writes it and I'm able to track down a link, I will of course share it here.
In the meantime, here is a database of known urns in this style: