The Lincoln Project comments on dog-murdering:

[Warpy]

6. Dogs are people pleasers

and at the bottom of a family hierarchy.

Also, Republicans are fear driven, so having a living alarm system that will bark and alert them to prowlers, fires, or other untoward events is comforting (Democrats like that part about dogs, too. The neighbors, not so much).

Other people keep them trained as hunting dogs, even if they never hunt.

Rural people need them to help herd animals and to raise a fuss if a predator shows up.

And for some Republicans, the dog is their best pal, Republicans are human, after all, no matter how much stupid shit they've been told and firmly believe.