She did not shoot.
She's the top dog in the top unit fighting against Kaijus.
She has her duty, she also has to protect herself, everyone around too.
Puting him in custody is already a risk when facing a Kaiju that can morph into a human.
Also, he's one of the greatest fortitude reading we've heard of. Not an easy choice.
And to top it all, his human form is her childhood friend.

Her decision, after she understands he protected all of them from death, gives her time to understand what happened and if she needs to eliminate our MC or continue the show.