There are a few assumptions here that might warrant challenging. Key amongst them the biggest...
Klingons have a warrior class, once even called a warrior caste. Is this the ruling class - or merely the class currently favored by the true rulers?
Generally, a warrior caste is generated out of a desire to keep idled warriors out of mischief, by people who want to see the warriors corralled up. These might be leading warriors themselves, or then other people of power, supposedly those in a position to pay or withhold pay for the warriors. There would be little inborn motivation for the warriors to isolate themselves from the society!
Fat cats like K'Mpec might let brainless warriors enjoy a preferred status of sorts as long as they also obeyed moronic "honor rules" that kept them from being a threat to the society. Scientists, farmers and hairdressers would complain (and not even under their breaths, because quite possibly every Klingon is
a brave one even if not a warrior) - but K'Mpec would fund them, too, to the degree required to make the Empire prosperous.
On occasion, a warrior would take charge of the Empire through palace coup. He'd soon be "corrupted" into becoming a sensible leader of a diverse Empire, though, only paying lip service to his earlier values. It is not as if any of the actual leaders we have seen would really have been handicapped by the stereotypical warrior thinking we learn from the likes of Worf.
The Empire overall would benefit from being bathed in warrior propaganda, because that provides not just stability but also patriotic comfort and looks good to the outside world. In times of internal trouble, stagnation and low moral, this might even take the form of religious fervor, with lots of Kahlessian mythology added on top of the basic honor code. But it would basically only affect the warrior class; others would keep on providing food, disruptors, starships and new Klingons to the Empire.
If it's all about the looks, then it's only natural that the audience may err into believing that warriors are the only thing of importance in the Empire. Our Starfleet heroes may make the same error.