Let's assume she fight it. What would realistically happen? She doesn't have enough money to save every orphan in the duchy.
And even if she had, she can't oppose the nobility directly. Not even the Archduke can do that, as evidenced later in the serie, even Sylvester has to be careful how he deals with the various noble factions to avoid beind potentially deposed. (part 5 spoiler)
If she tried to go directly against the nobility to impose her views, that would be akin to acting as the duchy ruler. She would promptly get a reality check from Sylvester and Ferdinand... Don't forget that Rozemyne's status is shaky at best. Sylvester can un-adopt her whenever he wants if she goes too far. And if she tried to go against Sylvester, even after the warnings, Ferdinand would certainly make sure she gets into a quick accident...
Rozemyne NEEDED the help of Sylvester to survive. On the other hand, Sylvester only needs Myne in order to enrich his duchy. If she begins to pose more problems than the money she might bring in later, then there will be no point keeping her.
As others have said, this story is not a power fantasy where you can do whatever you want just by wishing for it. Rozemyne's actions would have consequences, that could threaten her position or even her life, and the lives of her relatives as well.