I don't think this is like Cold Game, I think the prince is legitimately the prince and Jade is legitimately a knight.
They wouldn't send the prince in disguise to pick up some random girl from nowhere like Theresse. It would give her too much of an advantage. IF they were playing that game, he'd either go pick up all the girls (which is way too much) or the big hitters.
I think what will happen is the prince met her during her previous trip to the palace as a kid (that she forgot), and she played the flute for him or otherwise did something to make him fall in love with her. He's been searching for her ever since but realizes the noble girls he usually sees aren't that one magical girl, so this whole contest is his last ditch effort to figure out who she is. And when she plays the flute she'll give herself away.
Also, there has to be a reason why the other girls would bother warring with and competing with some random girl from nowhere who isn't even interested in winning. Well, if the prince suddenly made a move for her, she'd have every dagger aiming for her back out of nowhere.