What if Khan used the Genesis device strategically? Not to destroy Earth or another planet with billions of inhabitants, but he destroys a key strategic Klingon colony? Imagine a Starfleet ship enters the colony's space, Khan delivers the device, and away he goes. The scale of the destruction would be shocking, and the Klingons would not know it was a one-off attack. For all they know, it's just the beginning. They could logically think it was a surprise attack by the Federation and retaliate. So, Khan could possibly start a war that would be very costly in lives and resources across many planets.
If the Feds tried to explain, why would the Klingons believe any Federation explanation that the device was stolen and the attack an act of terror by an individual? In any case, they would blame the Federation and want retribution.