^^^ I'm pretty sure Cartwright was deeply in it by this point. Someone heard Kirk tell Spock "Let them [the Klingons] die" after everyone left the conference room, to filter it down to Valeris. A lowly Lieutenant would likely not have been allowed to come anywhere near that room full of top brass - but an Admiral would. And I doubt that was something that Spock would have told Valeris off-screen - too personal a moment between the two old friends. But it would be the perfect thing for a senior officer to use to manipulate a junior officer into doing "her patriotic duty" for the "greater good" of the Federation.
Also, Cartwright's odd, almost forlorn, expression on his face
when Kirk spoke up and said the Admiral was right, indicated to me that Cartwright knew exactly what was going on. He knew he was complicit in setting up one of Starfleet's greatest heroes (and the only one in the room to verbally agree with his assessment of the Klingons, augmenting his guilt over the plan) to be thrown into a faked attack on an enemy vessel flying a diplomatic flag, one or more murders, and a BOP that can fire when cloaked, sparking a massive interstellar war, swallowing the entire Alpha Quadrant, that could potentially kill millions, possibly billions. The fact that the Klingons had the time to prep an offensively-cloaking BOP for this mission means that certain members of Starfleet also had the time to set things up on their end.
Spock's mind meld on Valeris also basically implied that Cartwright was a part of the conspiracy from the get-go. There is no question, in my mind, that such was the case.