What kind of a man kills his brother?

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I agree. If a British soldier was willing to go against his country to help a hand full of Irish strangers escape from that very same prison, why couldn't Teddy do the same for his own brother? They accused Damien of being a coward at the beginning of the film, but in the end the real coward was Teddy.

I think you need to understand that this scene was done for dramatic effect. A lot of the incidents depicted in this film seem to have been based on actual events, such as the Kilmichael ambush, which was carried out by Commander Tom Barry's 3rd West Cork Flying Column, and the scene with the execution of the English landowner and his employee, which in real life was a woman, Mrs. Lindsey, and her chauffer, who were executed by a flying column commanded by Frank Busteed, after Lindsey learned of plans of another ambush which she reported to authorities, resulting in the capture and execution of six of Busteed's men.
The real tradgedy of the struggle depicted, was that after fighting shoulder to shoulder to gain their independence, many Irish, even families, found themselves facing each other in a brutal civil war which saw many atrocities committed on both sides against former comrades.
Although there is no record of one brother presiding over the execution of another, I believe this scene was written with the intent to highlight this tradgedy.

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It was for dramatic effect to symbolise that brother was killing brother in a struggle which echoes to this day. In reality he would have been relieved of this responsibility. No one would be so callous as to expect him to order the execution of his own brother.

You keep using that word. I do not think it means what you think it means.

Many supposedly did during the Irish Civil War (although I hear it was mostly close friends who fought shoulder to shoulder during the War of Independence and) even if it hurt them do so and probably haunted them for the rest of their lives. I can't imagine how it wouldn't.

I think that's what makes the film so tragic is, Damien in his letter tells Sinead to take care of Teddy as he's died a little inside and he still worries and loves his brother. But you can't blame Sinead either for her reaction to Teddy, and not wanting to see him ever again even if we saw Teddy tried everything he could not to do what he did and that he's a broken man himself from watching his own brother been shot. It's a terribly sad ending as Damien's death has lead to two people's lives been broken probably forever.