Just like Martok was driven by a hunger for glory and a desire to be perceived as greater than Martok during the Dominion War.
Since the Klingon Empire was the vanguard against Undine infiltration in the Alpha and Beta Quadrants, I would assume the Empire regularly verifies the identities of high-level, high-influence individuals.
Remember the Detapa Council's takeover of Cardassia way back when. The Empire didn't want or need proof that the changelings were involved with the power changing hands, it was only a broad excuse to wage war and take Cardassia over. The Klingons may be actively pursuing the Undine, but by and large it simply gives the Empire an excuse to conquer it's neighbors.
The Klingon-Cardassian war resulted in the Dominion gaining a foothold when the Cardassians willingly welcomed their aid in avenging themselves on the Klingons and retaking everything the Empire had captured. Shortly beforehand it had turned out that it had been the Klingons who had been infiltrated all along. And then Gowron turned out to be crazy, and more willing to die in a blaze of glory than to actually try to preserve his Empire.
The Klingon Empire refused to learn from history, and so doomed the Alpha Quadrant to repeat it.