Strictly speaking, there is no historical overlap of the Babylonian and Aztec empires by quite a margin. The generally agreed upon fall of the Babylonian Empire is dated to 539 BCE when Cyrus conquers Babylonia (1). The earliest dating for the Aztec empire would be the founding of Tenochtitlan in 1325 CE (2). Even if you want to be generous and consider the "Aztec Empire" to include the earliest Aztec identity, back when they were a nomadic people in northern Mexico, you can still only go as far back as 820 CE (3). Even then you have a 1350 year gap in between the existence of the empires.

My knowledge of the Inca is far more limited, and I rather wait for a more expert person to comment. However, my impression is that the Inca are considered a unique culture with no external influence before the arrival of the Spanish.

Reference:

1. Baker, H. D. (2012). The Neo-Babylonian Empire. A Companion to the Archaeology of the Ancient Near East, 2, 914-930.

2. Durán, Diego. The history of the Indies of New Spain. University of Oklahoma Press, 1994.

3. Bleichmar, Daniela. "History in pictures: translating the Codex Mendoza." Art History 38.4 (2015): 682-701.

As another ancient history nerd, no doubt it's a wild stretch to say that bronze age people were sailing around the world settling the Americas etc...Personally, I think these ideas have to be confronted head-on. Because if you look into the details of whatever "evidence" is given for all of the many Christo-centric historical conspiracies they are going to be false. I can only assume that there is some purported physical evidence, and this is where you can strike. Are Christian revisionist historians citing old sources and not giving its relevant context? Or are the objects cited simply forgeries, like "Hopewell mammoth pipes" and "Adena tablets with cuneiform" (which always seem to come up in their arguments).

Today, we underestimate the proliferation of 19th century pseudo-history in the USA...Mormons went so far as to create lots of rock art sites filled with mysterious symbols based on Egyptian, Mayan, and Anishinabe designs. These rock art sites became a part of the physical evidence Mormons (then and now) gave to prove their historical narrative. In reality, if you study rock art (which I have) it is immediately obvious the Mormon rock art was chipped into the surface very recently (the glyphs are very light, no weathered patina). Sometimes ancient glyphs happen to not develop a patina but what happened here is obvious...And rock art researchers have fully investigated it. So, citing their work (their papers are available online) is how we should answer their arguments.

Besides archeology, Christian conspiracies rely on a quite offensive tactic which is that they ignore oral histories. Which is to say, they claim their history is more true than histories given by elders from the descendants of the indigenous peoples in the Americas. While you and me both aren't elders privy to this information, it's obvious that those elders exist and their accounts are the most devastating rebuttal to all Christian pseudo-histories.

In 1875 Christians could conduct archeological digs and discuss the reports in church meetings...all without encountering an indigenous person. Today, they'd have to hire an archeologist and they could get someone like Lyle Balenquah: an archeologist and bearer of Hopi oral histories. He's said publicly that he knows that his mother's clan was at the Chaco Canyon town-pueblo which flourished ca. 800 years ago. But precisely how does he know this? Well, he won't tell you. Unless, are you a fully initiated member of his mother's clan kiva society? Oh no? Then you don't know that history.

Personally for me, that argument is something which Christian pseudo-historians don't encounter often enough. And they should, because pseudo-historians who write books and give talks at churches still to this day should never forget that if they attempted to talk about the real history of an elder's people to that elder's face, they'll get laughed at. In the documentary A Beautiful Mystery an archeologist asked a Pueblo elder "Are we getting at the right meaning behind this rock art?" The elder responded quote, "Don't worry, cause likely no matter what you say it won't actually be the right thing." Only people who don't know an elder would even try to speak over them when they told oral histories.

In this very long answer of mine, I gave lots of evidence that many societies today (and in the past) had their own conceptions of history. I hope with such answers that I can give support to people such as yourself, since so many people in the USA aren't aware of the massive extent to which Christian pseudo-history ignores literally everything else.

People have been arguing about Christian pseudo history for a long time. The Roman philosopher Porphyry argued that Christians and Jews who take the Book of Daniel seriously are ignoring its historical context. The text pretends to be an ancient composition with startling prophecies of the future many of which came true. But, it was obviously written during the 160's BCE when Judaea was under siege by the Seleucids. This is because its "prophecies" are just accurate history until the then-current events (the invasion); then the prophecies say that angelic armies of God are going to intervene and save everyone! It was common knowledge this didn't happen, as Porphyry was writing in the 300's CE. And Porphyry was correct, his interpretation of the historicity of the Book of Daniel is still agreed upon by modern scholars of Biblical Criticism. But if you sauntered into a Christian church's discussion group and said all this, then you too can rehash the same arguments Porphyry had with Christian apologists some 1700 years ago.

I've also written an answer to questions about the historicity of the Exodus narrative, so if you'd like to deeply investigate critical Biblical history then you'll enjoy the links I've attached to that answer.

And if you'd like to learn about the history of the ideology behind Jehova's Witnesses and other Adventist sects in the United States then you should start with this lecture by Alec Ryrie.

If you'd like to hear Lyle Balenquah talk about Hopi history then you can listen to him in this lecture.

If you'd like to rebut Christian arguments about Nephilim and giants, then you'd love Ancient Aliens Debunked which covers that question and so much more.

Pseudo-historical arguments always focus on explaining North American "mound-builders" and researchers such as Kenneth Feder and others have shown how these theories rely on 19th century race science and white supremacy.

People who collect North American historical artifacts (which is imo not ethical) encounter the problem of forgeries often enough that they feel the need to teach each other how to spot them, see this informative series of posts by user Painshill on an arrowhead collecting forum.

There are so many wacky theories about X Y Z ancient group who must have sailed to the Americas that you could spend a lifetime investigating and debunking all of them. In lieu of doing that, researchers such as Jason Colavito have debunked all those theories in this paper www.jasoncolavito.com/uploads/3/7/5/9/3759274/colavito_-_mysteries_of_ancient_north_america.pdf and the many others at his website.