The writing was a bit muddled in the first episode. I'm not entirely sure why firing first at the Klingons would do anything but get everybody on the Shinzou, as somebody mentioned above. Looking at it from a television production standpoint, it just seems like a cheap way to create drama and put Burnham where they need her to be. In universe, Burnham seems to have briefly lost her mind. And Sarek didn't do her any favors. Maybe she just imagined she was communicating with him, and he actually disdains her and only talks to her on Surakmas.