I'm not counting George Santos, because he merely created a GoFundMe for the dog of a homeless vet to get lifesaving surgery, and then took the money for himself.
https://www.cnn.com/2023/01/18/politics/george-santos-veteran-dog-gofundme/index.html

I'm not counting Mike Huckabee's son, because he wasn't a politician.
https://www.snopes.com/fact-check/huckabee-son-dog/

I'm only counting Kristi Noem once, even though she called for Joe Biden's dog to be killed.
https://www.nytimes.com/2024/05/05/us/politics/kristi-noem-biden-dog.html

I'm not counting Mitt Romney, since he merely terrified and endangered his dog when he put it in a carrier, strapped it to the roof of his car, and drove for 12 hours from Boston to Ontario. He had to stop once to hose off the dog's diarrhea dripping down the car's back window.
https://www.msn.com/en-us/news/politics/i-didn-t-shoot-my-dog-says-mitt-romney-after-some-compare-his-dog-scandal-with-kristi-noem-s/ar-AA1nZT4P

Update. I originally said the dog died in this incident. But the dog did not die.
https://en.wikipedia.org/wiki/Mitt_Romney_dog_incident

But I must update the count to include Todd Kincannon, once the general counsel and executive director of the South Carolina Republican Party. He fatally choked, stabbed, and mutilated his mother's dog in 2018, saying that God told him to.
https://archive.is/cYG87

I'm not counting Mike Collins (R-GA), since he merely hired a man to be his chief staff who was arrested for kicking a dog (animal cruelty) and whose criminal history had already forced him to resign as Trump’s Georgia state director. https://www.politico.com/news/2022/12/13/gop-operative-brandon-phillips-00073724

I should also count Mehmet Oz (aka Dr. Oz, Republican candidate for the Senate in 2022) who killed 300 dogs and hundreds of other animals in his academic research at Columbia University. On the one hand, his experiments were authorized by the university. On the other hand, they violated applicable rules and the US Department of Agriculture ordered Columbia pay a $2,000 fine.
https://www.jezebel.com/dr-oz-s-scientific-experiments-killed-over-300-dogs-e-1849609272

EDIT. Oz is now #Trump's nominee to run medicare and medicaid.
https://www.reuters.com/world/us/trump-picks-dr-oz-serve-cms-administrator-2024-11-19/

I'm not counting Rick Scott, who adopted a rescue dog from a shelter, named it Reagan, enjoyed some good publicity for that, and then returned the dog to the shelter. He and the shelter both refused to comment on the dog's fate.
https://www.tampabay.com/news/bizarre/after-2010-campaign-gov-rick-scott-gave-back-dog-reagan/1270497/

I'm not counting the Wyoming Republican party, which dominated the state legislature in 2021 and passed a bill to exterminate 90% of the state’s wolf population. The bill was so broad that it condoned much torture as well as direct killing.
https://www.outsideonline.com/outdoor-adventure/exploration-survival/wyoming-law-protects-wolf-torture/