That's technically true, yes. Louis doesn't actually have authority over her, and is 100% using her, though he is doing so more or less with the king's authority. The king gave him a mission, and the authority to carry it out using whatever means necessary, but it's not entirely clear whether that includes forcing other sages to cooperate. Louis may not even have seniority over her, given that they both were inducted at about the same time, and after she soundly defeated him.
The only reason she isn't being ordered directly by the king is because she's... Well, Monica. I'm pretty sure even the king doesn't know how to deal with her, or even whether she'd be capable of handling this kind of mission. She mostly just gets handed number crunching jobs, after all.
Either way, Louis giving her this mission to protect the prince is, itself, done as part of an order given to Louis directly by the king. Although technically, I don't believe Monica actually has to obey Louis here at all -- but she's too terrified of Louis to resist anyway.
The slaying of the black dragon was another mission Louis compelled (coerced?) her to do, which she also definitely didn't have to, but did anyway, because it's Louis and Louis is scary when he's mad.
All that said, the situation with Casey's punishment was the first time Monica actively showed defiance towards Louis, which was kind of a big deal, and he absolutely noticed.