First, we would have to have evidence that "the Sniper" did feel bad about killing people. I don't think that the evidence in the story supports this idea. "The head fell heavily on the turret wall. The woman darted toward the side street. The sniper fired again. The woman whirled round and fell with a shriek into the gutter." Here, "the Sniper" shoots the old woman as she runs across the street. Granted, she was an informant and had given his position away, but she is also an old woman and a non-combatant. Either way, "the Sniper" shows no hesitation or remorse as shooting either the man in the armored car or the woman.
Nor does he feel bad about having to kill the sniper on the other roof. He considers his opponent very coolly and in a very calculated fashion. "He must kill that enemy and he could not use his rifle. He had only a revolver to do it. Then he thought of a plan." He shows no hesitation here, either.
The one section of the story that supports his disgust with killing is this: "The lust of battle died in him. He became bitten by remorse. The sweat stood out in beads on his forehead. Weakened by his wound and the long summer day of fasting and watching on the roof, he revolted from the sight of the shattered mass of his dead enemy. His teeth chattered, he began to gibber to himself, cursing the war, cursing himself, cursing everybody. He looked at the smoking revolver in his hand, and with an oath he hurled it to the roof at his feet."
Now, this does imply that he feels bad about having to kill so many people. He is "revolted" by the sight of his dead enemy. He takes the opportunity to curse the war, himself, and everybody else. He seems to make an oath to "give it all up" when he throws his gun down.
Clearly, for this instant, he is fed up with killing. But I don't see it as a clearly expressed interior condition. Look what happens next: "The revolver went off with a concussion and the bullet whizzed past the sniper's head. He was frightened back to his senses by the shock. His nerves steadied. The cloud of fear scattered from his mind and he laughed. " He regains control of his emotions and chuckles to himself. He is once again under control. The lapse in his resolve was more likely caused by lack of food, water, and the fact that he is leaking blood.
I am not sure what you mean by "no one loved him when he was a child..." where are you getting that idea from? There doesn't seem to be much mention of "the Sniper's" background in the story.
Anyhow, best of luck to you!