For most of the story, the sniper feels very little regret at all. He coldly shoots two people below him before being shot in the arm himself. He didn't feel remorse then, and he didn't feel remorse for them later in the story either. The only person that the sniper kills that he feels regret for is the other sniper. Why would he feel this way?
One reason is revealed at the very end of the story. The sniper discovers that he shot and killed his own brother.
I think a second reason that he felt regret after having made the kill shot is that his adrenaline wore off. He had time to think about what he did. During the gunfight, it was all action and reaction.
The lust of battle died in him. He became bitten by remorse.
I think the third reason for his regret is that the sniper recognizes the skill in the other shooter. He regrets having to kill another man that shares his passion for accurate shooting and who is also at an equal skill level.
. . . he felt a sudden curiosity as to the identity of the enemy sniper whom he had killed. He decided that he was a good shot, whoever he was.