I haven't been given any reasons Admiral Marcus thinks he should start a war with the Klingons. That's why Section 31 is kind of farcical in STID. It makes no sense for them to go out of their way to start a war unless you're locked into an American government conspiracy allegory in which case it makes sense. This is one reason STID is bad - it sacrifices internal logic for the political allegory which isn't even a good political allegory.
Here's your reason direct from the horse's mouth. This is what Marcus told Kirk when Kirk came to tell him Harrison was on Kronos. It was just before he revealed to Kirk what Section 31 is and why it was created by Starfleet:
"All out war with the Klingons is inevitable, Mr. Kirk. If you ask me, it's already begun. Since we first learned of their existence, the Klingon Empire has conquered and occupied two planets that we know of and fired on our ships a half dozen times. They are coming our way."
In his mind, all out war was just an escalation of already existing hostilities between the Klingon Empire and the Federation, formal or not. Not everyone in Starfleet may have held that opinion, but enough did to support and get authorization for the Vengeance to be built. And since it was almost certainly black budget, if most in Starfleet who knew about it knew only one thing, they knew it wasn't being built as a ship of peace. He also knew he wouldn't get the Federation government to vote for a preemptive war, so he needed a pretense to justify one, however small. In his mind, he's preventing the day when Klingon cruisers show up out of nowhere in Earth orbit. Who knows if he's right or not? But the model of the Vengeance sitting on his desk shows that the ship is known to exist, so at least most of why it's being built must be aboveboard and others fear the Klingons and their aggression, too.
The only true secret he was keeping was that John Harrison is Khan.
He doesn't need a conspiracy of Starfleet officers and Federation Council members to start a war, only a pretense. Public opinion and necessity would take care of the rest. Khan's flight and Kirk's pursuit provided him the pretense for war. It was dropped right into his lap because Kirk came to him with the plan to get Khan, not the other way around. But when Kirk felt a sudden pang of justice and changed his plan, then figured things out ("Well, shit, you talked to him."), it became imperative for Marcus to destroy the Enterprise and its crew to cover up what he did to Khan, then spin that into an incident that could get him his war.