I think that It might be a mistake to dismiss the post-Columbian epidemics in the new world as being too slow.
They caused a massive population decline in both continents which lasted for centuries among the native - the Indian populations probably didn't stop declining and start recoving for about 400 years until about 1900. So the overall rate might be too slow for a spectacular event.
But some of the many, many individual epidemics which composed this continent wide centuries long population decline would have been very catastrophic.
The Mississippian culture was a Native American civilization that flourished in what is now the Midwestern, Eastern, and Southeastern United States from approximately 800 CE to 1600 CE, varying regionally. It was known for building large, earthen platform mounds, and often other shaped mounds as well.1 It was composed of a series of urban settlements and satellite villages linked together by loose trading networks.2 The largest city was Cahokia, believed to be a major religious center located in what is present-day southern Illinois.
The Mississippian way of life began to develop in the Mississippi River Valley (for which it is named). Cultures in the tributary Tennessee River Valley may have also begun to develop Mississippian characteristics at this point. Almost all dated Mississippian sites predate 1539–1540 (when Hernando de Soto explored the area),3 with notable exceptions being Natchez communities. These maintained Mississippian cultural practices into the 18th century.4
The Mississippian culture was a Native American civilization that flourished in what is now the Midwestern, Eastern, and Southeastern United States from approximately 800 CE to 1600 CE, varying regionally. It was known for building large, earthen platform mounds, and often other shaped mounds as well.1 It was composed of a series of urban settlements and satellite villages linked together by loose trading networks.2 The largest city was Cahokia, believed to be a major religious center located in what is present-day southern Illinois.
The Mississippian way of life began to develop in the Mississippi River Valley (for which it is named). Cultures in the tributary Tennessee River Valley may have also begun to develop Mississippian characteristics at this point. Almost all dated Mississippian sites predate 1539–1540 (when Hernando de Soto explored the area),3 with notable exceptions being Natchez communities. These maintained Mississippian cultural practices into the 18th century.4
https://en.wikipedia.org/wiki/Mississippian_culture
It is believed that pigs from De Soto's expedition introduced diseases which rapidly reduced the populations of Mississippian communities. The largest Mississippian communities would have been large enough for a few immortals to live as 1 in every thousand inhabitants.
The Cahokia Mounds State Historic Site /kəˈhoʊkiə/ (11 MS 2)1 is the site of a pre-Columbian Native American city (which existed c. 1050–1350 CE2) directly across the Mississippi River from modern St. Louis, Missouri. This historic park lies in south-western Illinois between East St. Louis and Collinsville.3 The park covers 2,200 acres (890 ha), or about 3.5 square miles (9 km2), and contains about 80 mounds, but the ancient city was much larger. At its apex around 1100 CE, the city covered about 6 square miles (16 km2) and included about 120 manmade earthen mounds in a wide range of sizes, shapes, and functions.4 At the apex of its population, Cahokia may have briefly exceeded contemporaneous London, which at that time was approximately 14,000–18,000.6
The population of Cahokia declined during the 13th century and it was abandoned by about 1350. So a nuclear family of immortals who found that theywere safe in Cahokia might have had to face death as the Cahokian population declined.
https://en.wikipedia.org/wiki/Cahokia#Decline_(13th_and_14th_centuries)
Similarly, early explorers report dense concentrations of villages along long stretches of the Amazon River, while later explorers reported the region was very thinly inhabited.
There were much larger cities in Mesoamerica and the Andes of South America. And Spanish colonists recorded various epidemics among the Indians in those cities in the centuries after the conquests of those regions. A specific epidemic which reduced the population of a large city by 25 percetn or 33 precent or 50 percent would have meant disaster to a group of immortals there who were close to the upper limit before the epidemic.
And of course there have been deadly epidemics in various old world cities too. Constantinople had a population of hundreds of thousands whentheplaguebrokeout in 541-549.
Procopius,[11] in a passage closely modelled on Thucydides, recorded that at its peak the plague was killing 10,000 people in Constantinople daily, but the accuracy of the figure is in question, and the true number will probably never be known. He noted that because there was no room to bury the dead, bodies were left stacked in the open. Funeral rites were often left unattended to, and the entire city smelled like the dead.[12]
https://en.wikipedia.org/wiki/Plague_of_Justinian#History
If during the plague over a thousand died each day for a year, the population of Constantinople would have declined by 365,000 duirng that year. And if the immortals in Constantinople were almost one in ever thusand, almost 365 immortals would have had to leave the city amoung the crowds of mortals that abandoned it. Each immortal would have had to select a destination town with a population of at least 2,000 to hid in, so that there would be enough people left over after the plague devastated that town, and hope that no other immortalwa s headed for that town.
So you need to look up the most populous cities in history, especially the most populous cities before the rise of modern medicine and sanitation, and then look up the major plagues which reduced their populations at avious eras.
What about the Chicago Fire in 1871? Comparatively few people were killed, but tens of thousands lost their homes and became refugees. And the Peshtigo. Wisconsin fire started on the same day and killed far more people, probably over 2,000, and thousands of others had to flee from the fire.
And what about the times when a large city with a vast population was captured by an enemy.
Vijayanagara was the large capital city of the Vijyananagara Empire in the 16th century.
The city was a powerful urban centre in South India from 14th to 16th century and one of the ten largest cities of the world.
An ongoing war between Muslim Sultanates and the Hindu Vijayanagara Empire led to the Battle of Talikota in 1565 CE, fought about 175 kilometres (109 mi) north. It resulted in the capture and beheading of Vijayanagara leader Aliya Rama Raya, mass confusion within the Vijayanagara forces and a shock defeat.[13][32][30] The Sultanate army then reached Vijayanagara, looted, destroyed and burnt it down to ruins over a period of several months. This is evidenced by the quantities of charcoal, the heat-cracked basements and burnt architectural pieces found by archaeologists in Vijayanagara region. The urban Vijayanagara was abandoned and remained in ruins ever since.[33][16][34] Vijayanagara never recovered from the ruins.[31][35]
The Italian Cesare Federici writing two years after the empire's defeat states that "The Citie of Bezeneger (Vijayanagara) is not altogether destroyed, yet the houses stand still, but emptie, and there is dwelling in them nothing, as is reported, but Tygres and other wild beasts."[36]
Sanjay Subrahmanyam states that Vijayanagara was arguably one of the only three centers during this period with a population of over 100,000 in South India and that from the contemporary accounts and what remains of its expanse, the city proper and the suburbs had a population of 500,000 to 600,000. He notes that Domingo Paes had estimated its size at 100,000 houses.[40]
So in the mere months of the Sack of Vijayanagara its population may have plummetted by over 500,000 as former inhabitants became fleeing refugees, slaves dragged away in chains, or rotting corpses. And if the immortal population had been close to the limit, over 500 immortals might have needed to find new homes in other cities or die.
And of course other cities may had even higher populations that Vijayanagara and been massacred by rutheless enemies.
The Mongols were notorious for massacuring the entire populations of cities they captured. In most cases those cities would have had only tens of thousands of inhabitats, room for only a small population of immortals.But a few had populations of hundreds of thousands. The Mongols also drove the rural population, much larger than that of the cities, to the cities, to labor for the Mongols in the sieges and provide human shields for attacking Mongols. And I expect that the country peopl wold also have been massacred after the Mongol victories.
So there were reportedly hundreds of thousands and even sometimes over a million persons reported killed at some Mongol sieges, though some historians consider those numbers exaggerated.
One very bloody day during the Mongol conquests was March 19, 1279, when the Battle of Yamen completed the Mongol conquest of Song Dynasty China. Thousands of Chinese soldiers were killed, but the majority of deaths were civilians, and at elast 100,000 are believed to have died on that day.
I note that possbily some immortals might find themselves in a large moving group of people, an army on the march, or a migrating horde of barbarians, or a stream of refugees. And if many of the others in the group died or were killed, the immortals might suddenly find that they were way more than 1 in a thousand.
Here is a link to a list of deadliest natural disasters.
https://en.wikipedia.org/wiki/List_of_natural_disasters_by_death_toll
So you need to find the bloodiest disasters and their dates. And then you need to find the lunar phases at the dates of those disasters. Since the "Moontime" when the immortals get slaughtered if their population percentage is too high is at the instant of fullest full moon, you need to find really big disasters that happened hours or minutes before the instant of full moon. You don't want a significant percentage to the deaths to happen after the full moon, but you don't want the deaths to be long enough before the full moon for the immortals to start to migrate to other highly populated regions.
And astronomy programs should be able to giv e the dates of the full moons closest to the various terrible natural disasters in history.
For example, here is a link to an online moon phase calculator.
https://everydaycalculation.com/moon-phase.php
And here are links to lists of bloodiest single days in military history:
https://militaryhistorynow.com/2014/11/12/bloodletting-the-deadliest-one-day-battles-in-military-history/
https://www.quora.com/What-has-been-the-single-deadliest-day-in-all-of-history
https://historum.com/threads/the-bloodiest-one-day-military-disasters.1104/
https://www.warhistoryonline.com/instant-articles/deadliest-single-days-in-war.html?chrome=1&A1c=1
The major bombings of cities include:
Dresden 13-15 February 1945, 25,000 dead.
Nagasaki 9 August 1945, 39,000 dead immediately.
Hamburg 24-30 July 1943, 42,600 dead.
Hiroshima 6 August 1945 50,000-60,000 killed immediately.
Tokyo 9-10 March 1945 80,000 to 130,000 killed.
And probably many thousands also fled from each of those cities, dropping their populations even more.
https://en.wikipedia.org/wiki/Civilian_casualties_of_strategic_bombing
The Battle of Cape Ecnomus in 256 BC might have been the largest naval battle in number of men present. The Romans allegedly lost 10,000 killed and the Carthaginians lost 30,000 to 40,000 killed and captured, making it a very bloody day and similar to the Battle of Salamis.
https://en.wikipedia.org/wiki/Battle_of_Cape_Ecnomus.
And psosibly you might want to consider changing the maximum proportion of immortals in a population so the immortals have higher population levels and so suffer more fatalities during moontimes following various disasters.
