Hmmm, on the one hand he's right in thinking that MC currently outranks him on paper but not in practice. Her relationship with the Marquis isn't great and it sounds like she doesn't have enough status/wealth to take over the household by force, so he might be able to win against her if he's built up a decent amount of credibility with the Marquis and plays his cards right.
On the other hand, if I were him, and I saw the Marquis bring in a wife from a not particularly impressive family sight unseen and then not really interact with her, I would think to myself, "Oh, he brought in a random lady to take over barely parenting his daughter."
Which, if that's true then said lady does solidly outrank me; she's basically there so that the Marquis doesn't have to deal with anything related to his kid and that includes me.
And even if all that is wrong I could lose this fight purely because replacing me is way easier than replacing her. Finding a new tutor = easy. Finding a third wife if he divorces MC = really hard.