Yes, in all the endings where the heroine doesn't end up with the prince or with a harem, the prince either dies or his status is unknown, or he ends up starting a war in his boredom, and in all of them the heroine's outcome also remains unknown. (By which I mean, she ends up with a capture target properly, but after that what happens gives off a dark undertone...like, as if they're going into their doom together.) So unless the Prince falls in love with her, the country and everyone, even the heroine, they are all basically doomed.
Those two so desperately believe that only the heroine character can save him is because when the prince and heroine kiss in the game, a special magical mark appears that doesn't appear for any of the other capture targets...hence the word "fated maiden", cause the game made it seem like only the heroine can make the prince fall for her, that only those two are meant for each other. And after falling in love he'll start being able to feel other emotions too rather than just boredom.
Bertia knows that the prince always pays lip service to his fiancee as custom so she can't believe he might actually be falling for her no matter what (and how could she if he himself doesn't know he's falling for her?). She won't tell the prince about these endings cause it'd be insulting him. Cause in the game the prince is unfeeling and uncaring, often called the android prince, so she didn't want to hurt his feelings and make him angry, saying he'll endanger the whole country because he's bored, because he couldn't attain one woman he'll do such a thing...he might get pissed and decide to stay away from their farce.
But no, she'll not die. She'll be exiled and made to atone for her sins as a nun devoting her body and soul to God...but the prince knew she wasn't really at fault here so he makes sure the punishment is actually something she'd want. (But her reasons for wanting it is gonna be kinda sad though, and that would be her own fault so it's even more sad for her, she won't be able to blame anyone but herself. The heroine deserves her own story, it's kinda bittersweet...mostly bitter and tragic.)