The Lincoln Project comments on dog-murdering:

[Shoonra]

13. American voters love dogs

Gratuitously killing a healthy dog simply rubs American voters the wrong way. Back in WW2, the Republicans tried to make points by criticizing FDR for having had a battle ship return to port where his dog, Fala, had been left by mistake. The Republicans thought the voters would react badly to turning the ship around, instead the voters rallied around FDR for retrieving his dog.

Frankly, the statistics show that more Americans shoot their wives than shoot their dogs.

Noem could have, obviously, put the dog up for adoption, found a new home for the dog, or even made better efforts to train this dog to hunt.