I don't believe the narrator of the story ever specifically states at what time Doodle becomes part of the family. The family had not expected Doodle to live, so he was never officially named. Doodle's father even had a carpenter build a tiny casket for when that time came. But Doodle did not die and after three months, his parents finally gave him the name of William Armstrong--making him an official part of the family. But Big Brother did not completely accept Doodle until after it was determined that Doodle was actually "all there."
It was bad enough having an invalid brother, but having one who was possibly not "all there" was unbearable.
Big Brother makes plans to kill him by smothering him with a pillow. But after Doodle gives him a grin, Big Brother ecstatically announces to the family that "He's all there!"