How might one formulate a constitution for a Non-Democratic Republic. Republics are often thought to be "the rule of law" as opposed to Democracies being "the rule of man". The latter is vote-dependent, the former is law-dependent, or dependent on the founding document. But the American Republic, despite having been quite dependent on the rule of law at its birth still had democratic institutions, such as representative legislative bodies.
How could one construct a Republican Government with no elected representatives? As Bastiat noted, the final result of democracy is a war of all against all. We see that today with teachers', police, workers, corporations', activists', and everyone else's lobbies. It's a country ruled by lobbyism. That is the necessary "conclusion" of a system of elected representatives. Since the political power of democracy is variable (as opposed to monarchies where it is relatively fixed), political ideology comes from without politics (from corporations, special interests, unions, media, etc.) because they are NOT variable. They do not have term limits. In the transition from monarchy to democracy, the source of political ideology is not transferred from the royal family to the people. It is transferred from the royal family to the economic royalty. The people are basically a sponge for political ideology, not a wellspring. Those with the most money, dedication and interest, then, are those who mold political ideology. This is why it is a huge mistake to think of today's political parties as "the voices of the people", because people are by and large "suggested what to think". Probably Chomsky does a better job of explaining this, but anyways I digress.
Do you have an idea of how a Non-Democratic Republic might function and how the rule of law, meaning those laws expressly declared at the start of the republic, might be adhered to with iron firmness as an intrinsic quality of the political structure itself?
The Anarch is to the Anarchist what the Monarch is to the Monarchist. -Ernst Jünger
The Articles of Confederation could be seen as a not very democratic Republic because only States could vote which meant it could never be the rule of men.
As you pointed out, the Constitution has individuals voting, while the Confederation had States vote.
Finally, the second Article of Confederation pretty much kept majoritarianism to a minimum.
Aren't China, Georgia, and North Korea examples of "non-democratic" republics?
The rule of law though, is an abstraction and as such cannot act. Only an individual can act and realize the rule of law in concrete terms. So ultimately the rule of law has to become the rule of a person.
So we should distinguish between the rule of a person per se from the rule of a person only insofar as he represents the laws. As such it is possible even for a monarch or dictator to rule only by the law. In the end, no system is infallible and so the only way to maintain the "rule of law" is for the leader, who is either born or appointed autocrat/consul/etc., to be a lover of the law he defends precisely insofar as it is the law.
As you say, the people are not the well-spring of ideology. So where is the source for our executor's motivation? It must come from a permanent code and the most permanent and inflexible is some variation of stoicism regarding the execution of laws. And who will teach this stoicism? Clearly an inflexible institution like an academy or a family tradition. I think it will be expedient also to limit the ideology only to a few people in order to purify it through conflict with more popular ideas, and through concentration/focus. But again, no system is infallible and what will make the teachers of our academy always so inflexible regarding their own teaching and who is to say that an outside force could not topple this gov or its counselors?