I've only found papers discussing Rome's population in the late republic and into the Imperial era.

That's because the only decent evidence we have is Republican. Roman history properly is really only from the fourth century, and even then those early events are hazy at best. Anything before that is a twisted mixture of legend and transmitted memory. The degree to which we can trust any part of the transmitted, largely Augustan, tradition is highly debated, but there is universal agreement that nothing before the fourth century should really be relied on too heavily.

The main source for the census is the Periochae of Livy, or Livy's actual text where extant. Our really good information from Livy is third century, beginning with the census of 294/3. From then on Livy gives us fairly regular information on the census, although there are a couple lustra that we're not sure about. The issues with the numbers in the Periochae are well known. The reading of the text is not entirely agreed upon, due to corruptions in the manuscript, and the figures are demographically impossible, changing by thousands of people up or down over the course of a single lustrum, which raises doubts about how well the census actually counted the citizen population. Moreover, there are disputes as to what the census actually counted, whether it included only male heads of households or all adult male citizens (or even women and children). The census was, like all state activities in antiquity, essentially voluntary. There was no way to coerce anyone into showing up to be registered in the modern nation-state sense. Rosenstein importantly argued that the variations in the census figures are often due to uneven reporting of families, particularly during times of crisis--otherwise Rosenstein agreed more or less with the low-counters, who are considered the majority, if not consensus, opinion.

This is all to say that Roman population data as it exists for us in the textual sources is all very disputed to begin with. We can say with some certainty that the census figures as reported present a general rise in the population from the early third century to the beginning of the first from around 250,000-400,000, but who exactly these figures are counting, their accuracy as reported, and the accuracy of the actual census are all very contested. Scheidel wants to get rid of them entirely, although his basis for replacing them does not seem very soundly based on any evidence. When we get any earlier than the third century it falls apart completely. Livy reports two earlier census, in 459 (3.24) and 465 (3.3) for which he gives the figures 117,319 and 104,714. That's rather a large jump in population, but not totally implausible, unlike some of the other census figures. Earlier than that we have to rely on Dionysius of Halicarnassus and Hieronymus, neither of whom we can rely on at all. Between the two of them we get figures for 508 (130,000), 503 (120,000), 498 (150,700), 493 (110,000), and 474 (103,000). These figures are plainly made up. They are not internally consistent, rising and falling massively. Moreover, these figures are absolutely enormous. Even if we suppose that these earlier census were counting all citizens--men, women, and children--despite the fact that the consensus is that that was not true of the later census, they're still impossible. We're looking at a city that's significantly larger than almost all cities in the Mediterranean at the same time, at precisely the period when we know archaeologically the city of Rome was not much more than a bunch of huts clustered in the Tiber valley. These data can be fairly straightforwardly discarded.

We simply have no idea how large the Roman population was during the kingship or the fifth century. We don't even know when the kingship lapsed--the traditional date is quite transparently an Augustan or late Republican borrowing from the expulsion of the Pisistratids at Athens. And even if the early figures preserve some shred of accuracy we really don't know what they'd be counting at this early date.