Impirren, your majesty. . . There is a timeline that I am working on which you are more aware of than most, one that I have had to do extensive research for(and I still have to do much more), and I would like to say that this is a topic I have given some serious thought. It was actually one of the first things I looked into.
The horses that were domesticated by ancient peoples 6000 years ago were much smaller than the horses of today, a little under three quarters of the size of today's horse. It, however, only took two thousand years to reach the current size, at which point breeding ideas took a very different turn, because now it could be ridden.
The modern llama is a lot less massive than the horse- the modern horse is five times as massive- but not much smaller in actual size, and it was bread to be a pack animal- it was not a very good pack animal by the time of Columbus, but it was a lot better than when it was first domesticated. Given an extra, oh say . . . 1000 years of heavy selective breeding, in a culture that is fairly well-connected(the inca have advanced roads, the ancient horse domesticaters did not), and the Llama will start coming close to the modern horse in size and will have possibly greater endurance, although still nowhere near its camel cousins from across the sea. At that point, it may finally be logical for a people to ride the llama for extended periods of time, and some of course would doubtless get the idea to bread them for stronger backs, so that by around, say, another 1000 years later, you'd have a fair analogue to the horse.
And now I will explicitly say that there's still another 698 years before the Spaniards come after that time elapses in the timeline you are helping do cartography for, enough time in my opinion to spread the llama beyond Peru.
EDIT: In hindsight I should've mentioned that the domestication of the llama would actually be like a combination of that of the donkey, camel, and horse, and of course would go faster if the Tawantinsuyu's infrastructure (not necessarily the empire itself, just the legacy it left) is not horribly disturbed.