The way I saw it, the prince only made her his knight and was getting chummy because she saved his life. Alex may have viewed him as a good friend, but I don't think he reciprocated
Probably doing it to save face, like how he pressured Alex to accepting the dance invitation. Could also be that it's far easier to dispose of a person who was already unfavourable in some of the public's eye rather than the person you just married for political reasons
Unless there's something I'm forgetting from the previous chapters, there's nothing stating that the galactic empire Princess comes from has the means to sufficiently retaliate
Since 'likeability' is important in this game, I wonder if vengeance also plays into that. People love to see characters get revenge on those who mistreated them after all.