I think one possible reason for why Doodle is so intent on burying the ibis is because he's a young boy that knows dead things are supposed to be buried. I'll briefly use some anecdotal evidence to support. I have three young children. We have a cat too. When the cat brings home a dead lizard or mouse, my kids have this need to bury it. I don't let them, but left to their own devices, my kids would bury the animal because they know dead things are supposed to be buried.
I think another possible reason that Doodle wants to bury the bird is because he has respect for the bird. Giving a proper burial to someone or something is a sign of respect. It's clear that Doodle is in awe of the bird and its beauty.
Doodle's hands were clasped at his throat, and I had never seen him stand still so long.
Doodle is mesmerized by the bird and its efforts to continue living. He's so mesmerized that Brother can hardly believe how still Doodle has become. When the bird dies, everybody just walks away. They don't see anything of value in the bird anymore. However, Doodle still sees great beauty in the dead bird. A proper burial will give tribute to that.
Even death did not mar its grace, for it lay on the earth like a broken vase of red flowers, and we stood around it, awed by its exotic beauty.
Finally, I think Doodle sees a bit of himself in the ibis. Doodle watches the ibis struggle and struggle to live even though it is in a broken state. The bird struggles to fly and do what birds do, but it just can't. It's too injured or tired, and it dies. For all of Doodle's life, he has struggled with his body not allowing him to do what a normal boy should be capable of doing. His own family expected him to die soon after birth. Doodle probably sees his own reflection in the bird, and he feels a great deal of compassion for the dead bird. That would explain why Doodle is so adamant that he properly bury the bird.