The Order of Yevon?
Aeons?
I am known throughout Spira by all but one; - Tidus
In the face of defeat and sorrow, I bring hope; - The Aeons are thought to be the only way to defeat Sin
Bestowed with valor, strong you will become. - The bless the summoner, and Yuna's faith is what makes her strong.
The song of songs, my reprise is a kaleidoscope - Hymn of the Fayth, and when an Aeon dies...
of the peace and prosperity of the rising sun. - the pyreflies look like a kaleidoscope held up to the sun.
It's the Hymn of the Faith
the first line refers to---as you mentioned---Tidus, as he's the only one who doesn't know the Hymn. The second line is my poetic way of saying that those who hear/sing it, feel stronger. You've nailed the third line As for the fourth, it's technically a continuation of the third, to finish my thought, that the hymn brings an air of hope and prosperity, as suggested by the sunrise.
So, I claim you got it, since you technically said the Hymn of the Faith
FFX
Chosen by many,
rejected by few.
I am the center
of this spiral of death.
I am in control,
control absolute.
Not Yunalesca either.
Hint: You're looking for a character.