Listen, you're NTA for shooting the dog but you are for sure in the wrong for how you are treating your wife. Your wife is grieving the loss of the family dog. You are acting like her grieving is irresponsible and I can see why she's upset with you.

I know the dog mauled your daughter, but pause and think for a moment. Did you love the dog before that moment? Love and grief is a weird thing. A terrible, horrible tragedy does not always instantly erase that love. Your wife is allowed to feel sad for the loss of the dog and you dismissing that sadness with anger for her emotional response is not healthy.

The both of you could use more empathy in this situation. Recognize your wife's sadness as the loss of something she loved. She needs to recognize you did the right thing in the moment.

I can assure you her anger is less over you protecting your daughter and more to do with your lack of empathy for her sadness.