Exact quotes from "Errand of Mercy". Anyone feel free to quote this post anywhere they'd like to in the future. Especially if it comes up in the SNW Forum, since I won't be over there.
1.
Kirk: "We've both guessed right. Negotiations with the Klingon Empire are on the verge of breaking down. Starfleet Command anticipates a surprised attack. We are to proceed to Organia and take whatever steps are necessary to prevent the Klingons from using it as a base."
Spock: "Strategically sound. Organia is the only Class M Planet in the disputed area. Ideally located for use by either side."
2.
Kirk: "Well, we've been anticipating an attack. I'd say what we've just experienced very nearly qualified."
Spock: "Yes, it would seem to be a very unfriendly act."
Uhura: "Automatic all-points relay from Starfleet Command, Captain. Code One."
Kirk: "Well, there it is. War. We didn't want it, but we've got it."
Spock: "Curious how often you Humans manage to obtain that which you do not want."
3.
Kirk: "Gentlemen, I have seen what the Klingons do to planets like yours. They are organized into vast slave labor camps. No freedoms whatsoever. Your goods will be confiscated. Hostages taken and killed. Your leaders confined. You'd be far better off on a penal colony. Infinitely better off."
4.
Kirk: "So, we're stranded here in the middle of a Klingon Occupation Army."
Spock: "So it would seem. Not a very pleasant prospect."
Kirk: "You have a gift for understatement, Mr. Spock. It's not a very pleasant prospect at all!"
5.
Kirk: "You have a lot to learn. And if I know the Klingons, you'll be learning it the hard way."
6.
Kor: "Captain of the USS Enterprise! A starship commander! And his First Officer? I had hoped to meet you in battle."
7.
Kor: "The fact is, Captain, I have a great admiration for your Starfleet. A remarkable instrument. And I must confess to a certain admiration for you."
8.
Kirk: "We have legitimate grievances against the Klingons. They've invaded our territory. Killed our citizens. They're openly aggressive. They've boasted they'll take over half the galaxy!"
Kor: "And why not?! We're the stronger! You've tried to hem us in. Cut off vital supplies. Strangle our trade. You've been asking for war!"
Kirk: "You're the ones who issued the ultimatum to withdraw from the disputed areas!"
Kor: "They're not disputed! They're clearly ours!"
The lines in that last section seem the most relevant to DSC. "They've invaded our territory. Killed our citizens." Fits Season 1, especially episodes like "The Butcher's Knife". The disputed areas can very well refer to all the space seized by the Klingons prior to "The War Without, The War Within". If it was as messy as I think it is, they might've stopped the war, but negotiations and disputes over every piece of territory could drag on for the entire 10 years leading up to "Errand of Mercy".
All the earlier lines make it sound like Kirk has first-hand experience with the Klingons. Whereas nothing Spock says indicates he does one way or the other. This fits with the Enterprise not being part of the Klingon War, but also still leaves wiggle room for SNW in case anything happens in the interim.