For all we know, the Empire did have an Emperor or a dozen during those 300 years; they were just bad enough to be erased from history. Conversely, there may have been several periods during which no Emperor reigned, or there were six simultaneously on the throne, and Klingon history glosses over those, too.
Not that we would need to believe in an Emperor during the specified 300 years, not in canon terms. But we do know the Klingon form of governing evolved somewhat during that time, as a woman could not serve in the Council during Azetbuhr's time, yet Gowron was offering a Council seat to K'Ehleyr a bit less than a century later, and didn't appear to be sarcastic in his choice of bribe. Probably the form of government is what the governors make it to be, as long as certain ancient customs are observed - and those customs might be more ancient than the form of government, and largely irrelevant to the day-to-day practices of governing.
Note that in "Sword of Kahless" we learn of the First Empire, and in "Til Death Do Us Part" of the Second Empire, in a way that suggests that Klingons have moved past that form of government, too. Perhaps these various Empires are separated by interregnums of which this three-century period is but one example, and the Kahless Clone has now inaugurated the Seventh Empire or whatnot.