To be entirely clear, and not risk miscommunication, Aire's goal with this plot is primarily to control the Lord-Commander's state of information. When Rele was forced to kill the Lord-Commander's spy, the obvious guess for the culprit of the murder was Christa. This was cemented when she vanished and Alard was framed for her murder the next day. The Lord-Commander certainly knows that Alard is not the type to feed a girl to his dragon. To the Lord-Commander, this will look like a relatively well-informed person deliberately moving against him by disabling one of his pawns. All of that suggests Christa had some rather interesting secrets, and was not who she appeared. This is, unfortunately, true, and very much not something they want the Lord-Commander investigating, lest his investigations lead him straight to Aire. Or at the very least to deducing that something took Christa's place after her return from her most recent trip. Aire is safest if nobody outside the Black Eyes even suspects she might exist.
The result that the Lord-Commander ends up pointed at the Red Queen was, initially, a happy accident; she doesn't know who she'd have chosen, if the news of the Red Queen hadn't rolled in that day, but she would have picked some group or another. The goal, when forming the plan, was simply to ensure that Lord-Commander Drystan would not continue investigating Christa in too much depth. If he thinks he has already solved the mysteries involved, he is less likely to keep investigating. The importance of that changed drastically when Christa's astrology responses came through and they learned that opposing the Red Queen is their holy mission, but it was not, initially, the primary goal.
Hmm. Perhaps even attempting to pin things on the Lord Inquisitor would not ring too false, as the culprits of the plot are supposedly relatively new fresh from the Pink, and so plausibly working on cliches themselves. But no, that's an option of last resort, better to have something that doesn't rely on the supposed Red Queen agents being incompetent.
Are there figures in the hierarchy of the Dragoons who would benefit from scandal falling on the Lord-Commander? Someone nearly well-positioned enough to make a bid for the position, but held back by the Lord-Commander's good reputation? This would not be enough on its own to change that, she thinks, but it could plausibly be the first step in a longer plan with those aims in mind.