Kirk was doing what every competent military commander would do in such a situation.
The Klingons were at war with the Federation. Kirk had a duty to impede their war fighting ability.
It's basic military training.
And what would be the cost of not acting? Unchallenged klingon control of Organia?
That would be out of character for both Kirk and Spock. They would, and did, fight the war until they were pulled out, dead, or it was no longer necessary.