The two “reasons” Orwell gives are 1) working elephants are extremely valuable; shooting one would be like “destroying a huge and costly piece of machinery” and 2) to avoid being laughed at by the Burmese in the town.
But there are other reasons for his reluctance to kill the elephant. Orwell is filled with disgust at his position as police officer; he hates the Empire he serves, he hates the townsfolk that ridicule him behind his back, and he hates himself for not having the courage to quit or fully embrace his duties as Constable. As he puts it,
With one part of my mind I thought of the British Raj as an unbreakable tyranny, as something clamped down, in saecula saeculorum, upon the will of prostrate peoples; with another part I thought that the greatest joy in the world would be to drive a bayonet into a Buddhist priest's guts.
The wild elephant episode becomes a example of Orwell’s predicament. He would much rather not kill the elephant, but feels helpless in the face of the large crowd that has followed him to where the elephant is, who clearly expect him to kill it. He feels compassion for the “coolie” that the elephant has trampled, but later expresses relief that the man had been killed, since it provided a legal justification for his actions. In either case, his true feelings about the matter are irrelevant. He realizes this at the same time he realizes he must kill the animal:
And it was at this moment, as I stood there with the rifle in my hands, that I first grasped the hollowness, the futility of the white man's dominion in the East. Here was I, the white man with his gun, standing in front of the unarmed native crowd--seemingly the leading actor of the piece; but in reality I was only an absurd puppet pushed to and fro by the will of those yellow faces behind. I perceived in this moment that when the white man turns tyrant it is his own freedom that he destroys.
The grisly and slow death of the elephant can be understood as a metaphor for the dying Empire, or Orwell’s own self respect. Either way, it is a kind of self inflicted wound, in which one's principles are sacrificed to save face.