I wish I could go deeper in a historiographical sense, but there's no evidence for this at all. That entire book is speculative neurobiology and is all based entirely on interpretations of the same literature all historians have access to. It's something that's virtually impossible to find falsifiable or to prove, and that's why even fifty odd years later there has been no form of verification for it. It remains purely speculative.

But it's an interesting example to cite specifically this very bold assertion of his, because: what's even harder than proving that humanity ever had a bicameral mind? Proving that some humans did, and others didn't, during a specific time period. This is where it's easy to get into some pretty nasty assertions about people, i.e. "they were inferior," and I feel it's rather irresponsible for Jaynes to take his pure speculation to such a specific, zoomed in example.

In short: you probably won't get any good answers because the source you're referencing is based not remotely on the kind of "minimal assumptions" form of understanding that any good historiographer will use. Instead, it's based on extrapolating an entire theory of the mind, just from literature. A good historian will say: we don't have even close to enough evidence to make that kind of assertion.

I'm not qualified to fully discuss Jaynes' hypothesis of the bicameral mind, because this is the first I've heard of it. A quick survey does not show it to be anything near an accepted idea in any of the academic fields it touches upon. So instead I went looking for what Jayne actually had to say about the Aztecs, their psychology, and the clash with the Spanish. I did not find much.

In Jaynes' book, The Origin of Consciousness in the Breakdown of the Bicameral Mind, he mentions the Aztecs only twice, both times in passing. The first time (p. 174) as an example of a bicameral society as evidenced by the Aztecs reporting their origin story involving directly speaking and being spoken to an idol. Jaynes places the Aztecs, in this case, in the same camp as the Old Testament Hebrews. The second mention (p.176) simply categorizes Aztec "picture codices" alongside Maya glyphs and pictograms at Uruk as forms of proto-writing.

As neither of those two examples comments on the Spanish Conquest, I kept looking. I did find a passage from The Julian Jaynes Collection which seems to more directly touch on the topic. During an interview, Jayne says:

The Aztecs, for example, had all the evidence of being bicameral at one time, following the voices, of their gods; but then the gods stopped speaking to them -- this is recounted in the Spanish records. After that time they turned to complicated methods of divination in order to find their gods, which is why they greeted Cortes with such enthusiasm. (p. 304)

Unless there is more to Jaynes' position on the Aztecs that my (admittedly cursory) search failed to turn up, it seems he is basing his conclusion on the idea that Cortes was welcomed as the literal return of the deity Quetzalcoatl. This is a debunked and frankly absurd myth predicated upon Indigenous people being childishly irrational, incapable of assessing facts outside of a lens of myth and superstition.

Actual scholarship on the Spanish Conquest exists which directly refutes the notion of a superstitious Aztecs quaking in their sandals in awe of a handful of godlike foreigners who then subjugated them handily. Townsend (2003) "Burying the White Gods" dissects how the notion of Cortes as the returned Quetzalcoatl developed as a post hoc rationalization of the Conquest. Restall (2003) Seven Myths of the Spanish Conquest challenges out-moded tropes of the Conquest, noting that the Spanish received absolutely vital help from Indigenous allies, including former members of the Aztec Triple Alliance, and that the "Conquest" did not fully erase Indigenous identity (this latter point vastly more explored by Lockhart's Nahuas After the Conquest wherein the author notes that Indigenous life essentially went unchanged for a generation of more after the surrender of Cuauhtemoc). Finally, Hassig (1994) retells the events of the Conquest while noting that all of the parties involved acted completely intelligibly and rationally given their own socio-political situations and incomplete information. Also, Hassig's text makes clear that the Conquest was not some quick singular battle, but a running series of clashes across the Valley of Mexico of a period of months, culminating in a months long siege of Tenochtitlan.

In other words, the premise of the Aztecs being conquered because they mistook Cortes as a god is not supported by the evidence. The actual history of the encounter between the Aztecs and the Spanish, in fact, militates against such a theory as sufficiently explanatory. By extension, Jaynes using the purported deification of Cortes as evidence for a society which is, or recently, was bicameral... well, garbage in, garbage out, I suppose.

Really, Jaynes falls into a long tradition of Western European scholars in a number of fields using the Aztecs as a foil to "Western Civilization." Sepulveda did this when he debated whether they were even capable of salvation, or were instead natural slaves. Humboldt did this when he stated the "feeble remains" of Indigenous works were evidence of their "intermediary place" in a linear hierarchy of human societies. the Harris and Harner did this when they posited a "cannibal kingdom." Victor David Hanson did this when he used the Conquest as an example of the superiority of the "Western way of war" (and by extension, Western civilization in general).

Psychology has not been exempt from armchair hypothesizing about the psyche of foreign peoples. There is a long, sad history of psychological scholarship being used to justify oppression and imperialism. Said's Orientalism is a classic work on how the construction of Western society is built upon contrasts with foreign (and strongly implied to be inferior) societies. More concisely, Okazai et al. (2008) note in "Colonialism and Psychology of Culture" note that

Colonial regimes are elaborated discursively by differentiating between the colonizer’s ‘superior’ or ‘more civilized’ ways of life and the colonized people’s allegedly ‘inferior’ or ‘savage’ ways. Scholars look to a wide range of domains to observe this subject-making: from medicine, to city planning, to exhibition, to ethnography, to science, to history writing – and of course to more obvious arenas of social control such as schools and the military.

Classing conquered people as intrinsically inferior thus opens up a tautology where anything they do is proof of their inferiority (which is assumed because they are conquered). Their architecture is primitive, their art savage, their language barbaric, their demeanor childlike, their minds unformed. As such, they are ripe for conquest and civilizing. Jaynes, knowingly or not, is contributing to this narrative and adding another brick to the great mausoleum of colonial dehumanization.