What specific advice would Atticus Finch from To Kill a Mockingbird give to Brother from The Scarlet Ibis, and how would Brother react to it?

In a lot of ways, Brother and Atticus are similar characters. They are both patient and hard-working individuals, and neither of them is willing to give up on a situation that the rest of the world sees as a lost cause. Brother refuses to give up on Doodle, and Atticus refuses to give up on Tom Robinson. I think the following quote is likely a piece of advice that Atticus would give Doodle.

"First of all," he said, "if you can learn a simple trick, Scout [Brother], you'll get along a lot better with all kinds of folks. You never really understand a person until you consider things from his point of view . . . until you climb into his skin and walk around in it."

Obviously, Atticus is telling Scout this pearl of wisdom, but it would also apply to Brother. By viewing things from Doodle's perspective, Brother would likely have further patience to handle Doodle's abilities and reactions to situations. Brother is already incredibly patient with Doodle, but he does every now and again fail to consider what Doodle might be feeling. This is most evident when they are running home, and Brother fails to recognize the terror in Doodle's cries. Brother runs off, and Doodle is tragically killed.
I do honestly think that Brother would take the advice quite well. It's advice that he would be craving from an adult since all of the other adults in his life seem to have more or less given up on what potential Doodle might have.