I’ve given it some thought, and I don’t think we should underestimate Elizabeth’s desperation. She’s been working towards becoming a hero her whole life, to the point I‘d doubt she would know what to do with herself if she wasn’t chosen. She’s been surrounded by people who told her it was impossible for a woman to be a hero, yet that only fuelled her desire to prove them all wrong. Then those two boys intentionally sabotaged her and when she tried to make her case, she was shot down. Now, while I will say the instructor was right to prevent her from participating while injured, his comment about a real hero not allowing themselves to be injured was not only unnecessary, it was basically saying it was her own fault for being injured and therefore she does not have what it takes to be a hero. I’ll add that someone with that line of thinking is certainly not going to punish the actual guilty parties so they still get to pretend they are hero material.
So keep in mind, Elizabeth was done. As far as she or anyone else knew, her chances of being a hero dropped to zero. Then along comes Beaubert who offers her a precious lifeline, one final chance to prove to herself and everyone she has what it takes. Not only that, she seemingly found one other man besides her father who believed she has what it takes. I don’t blame her for not looking a gift horse in the mouth.
She had one week. We don’t know how much time has passed, but it wouldn’t surprise me if it took a day or two to gather equipment and sneak out with it without someone sane stopping her. It’d take longer for her to carry all that with one arm, and these wanted criminals would have their hideout somewhere remote. Even so, she still would have had plenty of time for recon… if it weren’t for the fever.
Now obviously she isn’t going to give up what she thinks is her last chance to prove her worth as a hero, and she has no time to recover. Not to mention, it‘d be almost certain death if she passed out from a fever in an area where the only people likely to find her are the worst kind of criminals. So now it‘s a race, not against the one week deadline, but against her fever. She has to kill three men and make it back to the institute with proof she completed her task before the worst of her fever hits. And nowhere in that plan is room for further complications or delays. The time and energy she spent sneaking up on the first two men had probably already pushed her to her limit so even if Beaubart hadn’t set her up, she was probably doomed regardless.
The whole thing is beyond stupid and reckless. However, let’s not forget she was put in this impossible no-win situation by all those misogynists in the institute. Beaubert, Glorious, Kyle, even the instructors, they are all responsible for putting her in a position between giving up and letting them get away with their injustice against her (and any woman who aspires to be a hero), or continue to fight against increasingly unfavourable odds until she either succeeds or dies. Heck, even before her arm was broken, she admitted passing this final test was “do or die” for her, and evidently she wasn’t being hyperbolic. When she saw how hopelessly outnumbered she was, she didn't try to run, she attempted to take down her main target in a last ditch, suicidal attempt to prove her worth, if to no one else but herself.
In saying all that, seeing as she failed to live up to her own expectations, I can understand why she wasn’t thrilled to accept her appointment as hero right away.