I can think of two possibilities: (1) it is a mule supplied by the government, hence its ill-treatment is not of much concern to the person using it, or (2) it is a mule with the same slow work habits as other government employees, hence it needs to be beaten to make it do any work.
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"Rented mule" and "government mule" are two different idioms which have been combined in the question.
The prior answers adequately addressed the "rented mule" scenario: Abusing the animal because it (and mules are always "it"... they can't reproduce) is merely rented and not owned.
The "government mule" may refer to either 1) a mule owned/loaned by the government, perhaps as part