How does the sniper in The Sniper by Liam O'Flaherty betray his country?

The only thing that I can think of to answer your question is to say the sniper betrayed his own country because he was killing citizens of his own country.  The story never says what his total kill count is, but the story does make it clear that he kills an old woman, a man operating a turret, and the enemy sniper. Each of those victims is a citizen of the sniper's country, because the text says in the first paragraph that the Republicans and Free Staters were engaged in a civil war. On one hand I can see how being in a civil war is betraying your country, but I also know that both sides of the civil war don't see themselves as the betrayer. They see the enemy as the betrayer. So the sniper sees his brother as the man betraying his country, but the brother equally sees the main character sniper as the man betraying his country. 

There is perhaps another moment of temporary betrayal on the sniper's part.

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 curses the war and perhaps his role in it. But I don't see that as a full fledged betrayal because he takes no action against anybody or anything.   In fact two paragraphs later the sniper has decided to go back to his commanding officer and report in.