I can't shed light on the history of balloon animals, but I can, for posterity, address the Aztec angle. The source of the claim appears to be from a book by French magician, Jean Merlin, Great Balloons! : the complete book of balloon sculpting (originally publish in French as Entertaining Balloons in 1986). All claims of Aztec balloon animals come back to this book, specifically this ubiquitously quoted passage:

...the Aztecs were the very first people in history to make animals out of the bowels of cats to be presented to the gods as a sacrifice. The bowels were carefully cleaned, turned inside out, and sewn with a special vegetable thread whose main property was that it stuck to itself when left to dry in the sun, and this produced an almost airtight seal.

The bowels were then twisted and air was blown into them after each twist. When a particularly contagious disease exterminated most of the cats, they used the bowels of the corpses, and when these grew in short supply, human sacrifices were offered to the gods for the sole purpose of obtaining fresh bowels.

As Jacques Dupion Grouchard remarked in his beautiful book, The Mayannaise Connection, the simple making of one animal required several days. And there were only two models: the dog and a kind of donkey.

Once they had been made, the animals were carried (with great ceremony) to the top of the Aztec pyramid, where they were burned in praise of the sun... But it is only with the advent of rubber that the Mexicans began manufacturing balloons intended for modeling. The most famous among them was Senor Carlos, who was the first to come over to Europe to perform his balloon sculpting act at the famous Lido de Paris. Literally none of this is true.

The Mayannaise Connection (I can't even find out if this book exists) claim that that one of the animal shapes was a donkey doesn't work because, well, no donkeys. Unless we are talking about post-Contact, but then we have to speculate that highly visible rituals were being carried out when they would have been considered heretical, with all the attendant consequences.

A similar problem exists with the claim that cat viscera were used: no cats. Domesticated cats are a product of agriculture, specifically agriculture in Egypt/Southwest Asia, long millenia after the Bering landbridge was submerged. Unless we are positing that the "cat" intestines in question came from ocelots, we face the same problem as the donkeys.

I would also note that rubber had already done been advented some time before the Aztecs, as evidenced by the rubber balls we find in the El Manati excavation, dated more than 3000 years ago. I'm going to assume this is more of a translation issue though, and Merlin is referring to the invention of modern thin rubber balloons. This would make sense, as the "Senor Carlos" mentioned only passed away last month in Connecticut, far from his birthplace in the exotic Mexican state of Massachusetts.

The note about harvesting organs from dead bodies, or even committing sacrifices specifically to obtain bladders and intestines and what not, is the most ridiculous part. That is not how Aztec funerary rites worked and it is certainly not how sacrifice worked. Sacrifices were made for specific religious and political reasons, not to pull out random organs for the purpose of creating balloon animals, of which there is no mention in either Sahagún's General History, Book 2: The Ceremonies or Durán's Book of the Gods and Rites, which I have now spent far too long searching through for anything resembling an inflated intestine.

I would chalk the purported Aztec connection up to a magician's flim-flammery, using an "exotic" connection to the past to embellish the lineage of a modern practice.