The Judge
In this relationship, the family dog plays the judge, jury, and executioner roles. The enthusiasm to win its favor begs the question: Does the poor pup long for a peaceful walk, not marital theatrics?
All jests aside, it might be in the OP's and his wife's best interest to discontinue this practice. Though it may seem harmless and amusing, indulging in partner disparagement isn't a healthy habit to cultivate.