Ooh, this is a deep-cut request. It's less about a literal princess and more about a daughter raised in that gilded cage, forced to navigate its brutal politics. A classic that comes to mind is 'The Maddest Obsession' by Danielle Lori. Gianna isn't just ornamental; she's smart, observant, and uses her position and wit as her primary weapons against men who see her as a prize. Her 'dangerous rivals' are often the very suitors and allies within the organization, and the tension comes from her maneuvering within those constraints.
For something with a more direct, hands-on approach, I'd look at 'Reaper' by A. Zavarelli. The heroine isn't just surviving; she's actively training and preparing to take on a rival faction that wronged her family. It's grittier, with a stronger revenge plot driving the 'overcoming' aspect. The danger feels more immediate and physical compared to the psychological chess game in some other books.
A slightly different angle is 'Crow' by A. Zavarelli, where the heroine is essentially bred as a 'mafia princess' but her entire life is a setup by a rival. Her overcoming involves untangling that web of betrayal from the inside, which adds a fantastic layer of paranoia and suspense to the dynamic.