Honestly, I assume the only reason she's not saying yes to such an enticing offer of being the male lead's lover is because (well, not counting 1. having her life in danger since the ML will naturally have many enemies and 2. his blood soaked history/image and the obvious 3. No interest in him currently whatsoever) back at the early chapters, she assumes fate will find a way to force the plot to continue (this is what she assumed would happen to the letter if she didn't send it) and he'll most likely have a hand in her execution.
Still, weighing the pros and cons, she should just say yes. She gets to be with the ML and will be co-in charge of his organization, ML is especially a beast in bed, is incredibly handsome, humurous, a bit vain though (Chapter 16 last dialogue), if she stays in good graces with him, he has no reason to execute her (heck, her actions of avoiding him could've given him an excuse to punish her despite being a princess since she doesn't have any power). Most likely lil tsun adorable brother won't deny the relationship between his favorite sister and his idol/hero. Most importantly, she can rub it on Edith's unbearable *ss (I wanna say unreasonable but someone mentioned Edith's doing all this harassment since MC is emperor's favorite, barely has any training or anything to do at all and yet they share the title of princess, all her training went kapoof since MCs existence could take what's left of her value as a princess away)
Oh? We also got to see Lancelot, the adorable sweetheart has 2nd ML written all over him