Adopt The Saudi Arabian Pattern Of Succession
A Large Pool Of Potential Future Kings Related To The Dynasty Founder
In the Saudi Arabian succession system, all adult male descendants of the founder of the dynasty are eligible to become king someday, and there are lots and lots of descendants (in the Saudi case, due to a combination of polygyny and successive marriages and high fertility for nobles). There are basically basically no "cadet" lines with only a remote chance of succession if lots of unlikely deaths occur no matter how distinguished their own efforts may be.
The King Has Great Freedom To Hire And Fire Royals From Powerful Posts
In the English hereditary monarchy, most of the sub-posts were also hereditary under a primogenitor system, creating many points where incompetent aristocrats could weaken the kingdom. This is not true in the Saudi system.
The sitting monarch can name a Crown Prince and other senior cabinet members from the most trusted, capable and competent of his adult descendants whom he interacts with regularly.
Less trusted, less capable and less competent descendants of the founder of the nation are given posts commensurate with their abilities. One might be given charge of a minor government ministry, another a governorship of a province or a mayorship of a city or village, or an officership in the military as high as the descendant can handle, or a chancellorship of a college. Black sheep might be given an allowance but no responsibilities at all - but denying them any benefits of their royal blood could breed dissent.
All lesser officials would serve at the king's pleasure, rather than for life, creating stronger incentives, allowing the king to rotate promising candidates through different parts of the kingdom and different experiences to become qualified, while also preventing any royal from consolidating too much power that could be mobilized against him.
Succession Is Decided By A Council Of Senior Royals
A council of royals who are senior officials in the regime, like a board of directors, would have formal power to appoint a successor king without regard to whether or not that person was Crown Prince, although the Crown Prince would be favored. The size of this council would vary over time based upon the monarch's wishes.
If the Crown Prince position is vacant when the monarch dies, or the Crown Prince has lost support from his kin, the senior members of the ranks of potential heirs choose someone, usually one of their number, to be the new monarch.
If the sitting monarch has chosen wisely, the most qualified member of the royal family will already have been named as Crown Prince and the succession will be seamless.
But, if a demented old man who named a flatterer with a rotten heart to be Crown Prince, the empire is not lost. And, because no one has a true right to be the next king, even the Crown Prince, it will be hard for a disappointed contestant to mount a contest to the pick of the senior eligible heirs and unwise when the disappointed contestants are likely to get some of the most senior posts in the new regime.
Benefits Of This System For Francia
This still provides some of the key benefits of a monarchy to a medieval state:
(1) if you don't have the resources to groom even 1% of the population with the training and opportunities necessary to prepare them to lead the country, by limiting scarce training and education resources to descendants of the founder, you can get a qualified leadership elite (e.g. mostly literate and training in riding horses and war) with a minimum of expenditure of resources in a desperately poor society.
(2) it provides clear lines of authority direct from the king to organize and mobilize the country which fits with the very direct managerial authority that Charlemagne exercised over his vassals in the early kingdom - with a traveling court that regularly visited every place in the domain; he could bring unassigned royals in his entourage to be dispatched to new posts as the need arose in the circuit of his traveling court, replacing incompetent officials who were discovered en route,
(3) it provides for rapid succession that is not subject to dispute and provides a clear outcome (often monarchy is justified on the ground that avoiding civil wars of succession created by uncertainty is more important than the quality of the king), and
(4) it would protect the nation from the mishaps of a purely hereditary succession system with no regard to merit. The selection of a monarch and senior officials based upon merit within the class of descendants of the founder avoids the elevation on monarchs like Charles the Fat who could end a dynasty.
This system also creates a way to reward and incentivize otherwise lazy aristocrats to perform in order to gain more power.
Since 98% of them would have no shot at real prominence and power in the European style primogeniture system, there is a large corps of powerful people who are better trained and educated than anyone else in the kingdom to have a vested institutional interest in supporting this alternative approach and the monarch it selects, even if they don't personally get the top jobs.
Those who come out on the bottom in this quasi-merit based system are the least qualified to mount a regime changing coup.
It also divides up power, in so many ways, among so many people, that no one descendant can consolidate enough power to split the empire and create an independent region if it is managed well by the monarch.
Family Dynamics In A Francia's Version Of This System
In Francia, it wouldn't be necessary to have formal polygyny.
It would be enough to abolish the succession distinction between legitimate and illegitimate descendants of the founder of the nation to get a sufficient pool of eligible descendants to find a few worthy of responsible positions in the empire. Royals would have no trouble producing many descendants if their children, marital and non-marital alike, had such promising futures.
Also, if a woman lied and said her child had a royal father when the father was actually not royal, and was believed, there would be little harm to the kingdom - if the child was competent he would serve the empire well; if not, the cost would be little more than the cost of a modern society to reward the occasional lottery winner who doesn't really deserve the prize. Relying on the royal parent's declaration to determine paternity while ignoring commoner testimony would encourage people to try to win the favor of royals they were in close relationships with, lest their children be disowned.
This system would also provide a means by which promising non-royals could join the dynasty, marrying (or merely mating) a royal family member and having children with them who would be royal descendants too. These marriages might be subject to approval of the monarch, and would provide a means by which the empire could co-opt anybody who might otherwise lead resistance against the regime.
How Would The System Be Imagined?
The system could be implemented by Charlemagne, perhaps consciously adopting pre-House of Saud Bedouin traditional leadership succession practices he learns about from spies and captives in his war with the Moors (figuring he must copy the techniques that give them an advantage, while adapting them to his own world), and followed by his successors, perhaps with the support of a prescient Pope who sees the perils of the alternative that was actually followed in our world.
Other examples of similar systems might also suggest it.
One would be the Roman Catholic Church's College of Cardinals (a group of Bishops hand picked by the Pope to vote on a successor to him).
Some nominally democratic Northern Italian and Swiss cities states called communes operated on something reasonably similar to this basis from the late middle ages until the 19th century. As Wikipedia explains:
During the 11th century in northern Italy a new political and social
structure emerged and the medieval communes developed to the form of
city states. The civic culture which arose from these urbs was
remarkable. In most places where communes arose (e.g. France, Britain
and Flanders) they were absorbed by the monarchical state as it
emerged.
Almost uniquely, they survived in northern and central Italy to become
independent and powerful city-states. The breakaway from their feudal
overlords by these communes occurred in the late 12th century and 13th
century, during the Investiture Controversy between the Pope and the
Holy Roman Emperor: Milan led the Lombard cities against the Holy
Roman Emperors and defeated them, gaining independence (battles of
Legnano, 1176, and Parma, 1248 - see Lombard League). Meanwhile the
Republic of Venice, Pisa and Genoa were able to conquer their naval
empires on the Mediterranean sea (in 1204 Venice conquered one-fourth
of Byzantine Empire in the Fourth Crusade). Cities such as Parma,
Ferrara, Verona, Padua, Lucca, Mantua and others were able to create
stable states at the expenses of their neighbors, some of which lasted
until modern times.
In southern and insular Italy, autonomous communes were rarer, Sassari
in Sardinia being one example.
In the Holy Roman Empire, the emperors always had to face struggles
with other powerful players: the land princes on the one hand, but
also the cities and communes on the other hand. The emperors thus
invariably fought political (not always military) battles to
strengthen their position and that of the imperial monarchy. In the
Golden Bull of 1356, emperor Charles IV outlawed any conjurationes,
confederationes, and conspirationes, meaning in particular the city
alliances (Städtebünde), but also the rural communal leagues that had
sprung up. Most Städtebünde were subsequently dissolved, sometimes
forcibly, and where refounded, their political influence was much
reduced.
Nevertheless some of this communes (as Frankfurt, Nuremberg, Hamburg)
were able to survive in Germany for centuries and became almost
independent city-states vassals to the Holy Roman Emperors (see Free
imperial city).
In the later stage of the commune system, they started to resemble the political system of the House of Saud. Council members generally served for life and only aristocrats had any political say. Per Wikipedia:
During the 17th century seats in the councils became increasingly
hereditary. There were between 50 and 200 families that controlled all
the key political, military and industrial positions in Switzerland.
In Bern out of 360 burgher families only 69 still had any power and
could be elected by the end of the 18th century. However, the
aristocracy remained generally open and in some cities new families
were accepted if they were successful and rich enough.
A less likely contemporaneous example, but not impossible for him to learn about would be a Hindu joint family:
Historically, for generations India had a prevailing tradition of the
joint family system or undivided family. The system is an extended
family arrangement prevalent throughout the Indian subcontinent,
particularly in India, consisting of many generations living in the
same home, all bound by the common relationship. A patrilineal
joint family consists of an older man and his wife, his sons and
daughters and his grandchildren from his sons and daughters.
The family is headed by a karta, usually the oldest male, who makes
decisions on economic and social matters on behalf of the entire
family. The patriarch's wife generally exerts control over the
household and minor religious practices and often wields considerable
influence in domestic matters. Family income flows into a common pool,
from which resources are drawn to meet the needs of all members, which
are regulated by the heads of the family.
There was trade between Italian states and India during this time period, so it is not entirely far fetched that Charlemagne might learn about this system.
Another more familiar frame for the concept (although unlikely to be an inspiration for Charlemange) would be a family owned business corporation in which all family members in the bloodline own shares that provide them a modest stream of dividends, the CEO engages in lots of nepotism to hire family members for key posts, and a board of directors (de facto appointed by the CEO from the ranks of adult shareholders) chooses a successor CEO, instead of having his CEO post pass by intestate succession (which is essentially what the European hereditary model involves).