Other mistake: Ambrose smacks a stone on Ethan's forehead. No blood nor even a bruise appears seconds later when Ethan is kicked in the face and we get a close-up of him.
Other mistake: Luther tells Ethan that he will only have 40 seconds for the louvres to remain open for him enter the ventilation shaft at Biocyte before the alarm goes off. But the warning message on Luther's computer says that the limit is 20 seconds. (01:06:30 - 01:09:10)
Answer: Ethan did not want her to kill herself. Giving her a gun would allow her to do so.