Burke is saying here that there’s a natural order even in society, in that some classes are meant to rule and others are supposed to be ruled, and when the rebels seek to “level” the social classes, they “war with Nature.”
This seems to me a misreading of Burke. Class, as I understand it anyway, is inherited. Yet Burke refers to people by their occupations. He denigrates (as governors) the tallow-chandler, but not the son of a tallow-chandler who becomes a prosperous member of the non-nobility. I see Burke's "natural order" as relating to talent, not birth.
The fundamental problem of governance, agonizingly labored over in The Federalist, is how to protect the political interests of the inmates in the asylum without allowing them to run the asylum. Therein lies the difference between the French and American revolutions. There was no French Madison, Jay or Hamilton. Quel domage.
For me, of course, there is no such thing as "natural law." There are, however, some plus-sum strategies that work because they reject defection in that massively multiplayer iterated prisoner's dilemma called life. Natural law is the opposite and antidote to Hobbes's "natural state." Making up stories to avoid learning game theory is a time-honored tradition we should reject with "infinite caution."