Since Barry's death was relatively recent (1920), I was surprised to find this song in Carl Sandburg's American Songbag (1927). The book has several verses I haven't found elsewhere.
-Joe Offer- Sandburg's notes:

Tongues of love and hate, breaths of passion and suffering, all mingled with a strange bittersweet, are in this song out of the violent events in Ireland. Probably all wars and revolutions produce figures like Kevin Barry, though seldom do they have such adequate songs as memorials. In Nashville, Tennessee, one may look at the statue of Sam Davis, who died refusing to turn informer and thus save his life. Davis has a statue in bronze; Kevin Barry has a song. These verses and their wistful, longing melody are from Irish boys and girls in Chicago who learned the ballad on the Ould Sod.

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KEVIN BARRY

1. Early on a Monday morning,
High upon the gallows tree,
Kevin Barry gave his young life
For the cause of liberty.

2. Only a lad of eighteen summers,
Still there's no one can deny,
As he walked to death that morning
Nobly held his head up high.

3. Another martyr for old Ireland,
Another murder for the crown,
Brutal laws to crush the Irish
Could not keep their spirits down.

4. Lads like Barry are no cowards.
From their foes they do not fly;
For their bravery always has been
Ireland's cause to live or die.

5. "Kevin Barry, do not leave us,
On the scaffold you must die!"
Cried his broken-hearted mother
As she bade her son good-bye.

6. Kevin turned to her in silence
Saying, "Mother, do not weep,
For it's all for dear old Ireland
And it's all for freedom's sake."

7. Just before he faced the hangman
In his lonely prison cell,
The Black and Tans tortured Barry,
Just because he wouldn't tell

8. The names of his brave comrades,
And other things they wished to know.
"Turn informer and we'll free you."
But Kevin proudly answered "No!

9. "Shoot me like a soldier.
Do not hang me like a dog,
For I fought to free old Ireland
On that still September morn.

10. "All around the little bakery
Where we fought them hand to hand,
Shoot me like a brave soldier,
For I fought for Ireland."