Tribbles are vermin on any world without native species that will prey on them and will on such worlds cause ecological disasters.
Tribbles and Klingons are incompatible as species.
Even Kirk didn't want Tribbles on his ship.
Ecoterrorist Montgomery Scott was fully aware of these facts. As was First Officer Spock who collaborated in the terrorism.
Finally it was the TNG Bonehead Klingons not the TOS Klingons who considered vermin extermination to be a war. It shows how the Federation genetic warfare damaged the Klingon race dragging them down to the human level. In any case the vermin infested Tribble home world was used as a source of terrorist weapons by the Federation scum either it had to go or the Federation did. Removing the weapon source was more cost effective.
Finally remember Kor beat Kirk. It took demigods to turn things into a draw.