They should keep the monarchy, but find a really charismatic royal dog to appoint as monarch instead of Charles or William. All royals really need to do is be good at shaking loads of hands, greeting lots of people, and posing for photos, and dogs are better at all of that than people. The Queen is 94! She shouldn't have to shake a hundred people's hands all in a row.
Dogs have large litters and can have multiple litters more easily than people, so there'd be constant pregnancies and new babies for the press and the public to coo over, without the insane cost and uncertain future of non-heir royal human babies, and without getting into the thorny issue of royals whining about the environmental cost of over-population while having large families themselves.
Dogs are also good at causing minor scandal so the tabloids would be happy ("King Snuffles caught sniffing other dog's bottom!") but would never cause any really serious scandals like financial fraud or being mates with pedophiles.
Dogs stay out of politics.
Most people like dogs and dogs are far less controversial than royals.
Yes. Snuffles for King 2021.