It's safe to assume that virtually everything Rose engages in is due to the Duke allowing it to happen. The duke isn't necessarily trying to control her every step so much as he is trying to influence her thoughts and direct her actions to his benefit. So if Rose decides to leave the manor, sneak off to the slums, work a shift at that dive bar, and turn a few tricks then it's to the duke's benefit because he wouldn't allow it if it wasn't. The duke gives the impression of being a power mad control freak, but he doesn't really seem to care if his guards or other nobles fuck members of his maid harem. It's well within his established personality to let Rose to go out and have fun so long as she doesn't do anything that endangers herself or his power over her.