theory

Give me a bit to write it up and I'll send you what I got!

But, I wanted comment on something you mentioned before I go.

... but I'm confused by the static part. Naively, someone might interpret the phrase "prices that allow an economy to reproduce itself" to mean "prices that cover production cost plus owner spending habits," because then everyone in the economy breaks even in their expenses and earnings over time, and the economy remains static.

I don't think that's a naïve interpretation, and in fact I think the insight there is spot on! When I say that natural prices are those prices that allow for the system to reproduce itself, I do mean the capitalist system. So natural prices under a capitalist formation must be able to reproduce the class exploitation of the capitalists (and hence their consumption), else it isn’t capitalism that is being reproduced.

In fact, bringing this insight into a definition of value is something that Ian Wright has done, and is part of his solution to the transformation problem, i.e. how to get natural prices from value. Essentially, his argument as I understand it is that one can create an extended measure of value that includes the labor required to cover products made for the owner’s consumption (surplus value), and that this extended measure of value is still a measure of value, just through lens of your insight. This extended measure of value is still measured in labor-hours, and is commensurate with prices under capitalism because while the surplus labor is not necessary for the workers, it is necessary for the system of capitalism to reproduce itself.

Though, like I mentioned before the transformation problem is worth its own post. And this is one solution, of many proposed, for the transformation problem.

The rest of your question was on the uniqueness of the solution

the problem would have no unique solution—there are infinitely many ways to assign arbitrary wages and prices in an economy

and I'll respond with a write up on solving for natural prices with the framework I'm most familiar with.