What does the scarlet ibis symbolize?

The scarlet ibis is symbolic of Doodle himself.  First, the scarlet ibis is not native to the area.  It is native to the tropics.  

"It lives in the tropics-South America to Florida. A storm must have brought it here."

So what you have is a bird that is completely out of place in its current surroundings.  That is a lot like Doodle.  He is small, fragile, and handicapped.  Brother tries to incorporate Doodle in...

The scarlet ibis is symbolic of Doodle himself.  First, the scarlet ibis is not native to the area.  It is native to the tropics.  

"It lives in the tropics-South America to Florida. A storm must have brought it here."

So what you have is a bird that is completely out of place in its current surroundings.  That is a lot like Doodle.  He is small, fragile, and handicapped.  Brother tries to incorporate Doodle in all kinds of things, but Doodle just can't quite keep up in the world that Brother demands.  The bird too is small and fragile.  It too is injured and broken in a way that prevents it from surviving in the world that it finds itself in.  

At that moment the bird began to flutter, but the wings were uncoordinated, and amid much flapping and a spray of flying feathers, it tumbled down, bumping through the limbs of the bleeding tree and landing at our feet with a thud.

I also see the scarlet ibis death scene as symbolic foreshadowing of Doodle's death.  There are a lot of similarities between the two.  Both were essentially killed by a storm.  Both died underneath a red plant, and both are covered in red.  The bird is naturally red, and Doodle is covered in blood.  

He had been bleeding from the mouth, and his neck and the front of his shirt were stained a brilliant red.