I understand the frustration with a stupid FL, but this one doesn't deserve it. If anything I find her admirable for having the courage to kill herself.
If you look at ch 1 again she took the poison AFTER the ML said some stupid inconsiderate things, like imply her feelings don't matter or that she's only good for baby making and not as a partner. (Key word is imply based on the interaction). Sure she had it there beforehand, but she took it once she felt truly hopeless regarding her future. Otherwise she would have taken it earlier.
The ML was in the wrong. He could have easily explained he loved her and wanted to help her kingdom or something, but he just says she and the country are his, showing a ruthlessness and that he didn't care for her. It's not just the rumors. Its his very actions that fed a bad reputation. Plus he bedded her without taking the time to explain to her his affections or that he cares for her.