It doesn't really make sense to shift this particular sentence into the past in this way. If you're going to use the past perfect here, you need to mention a period of time, such as "when he had been a child for ten years". Suppose we change the sentence a bit:
He said that after he had been out of school for five years he would get up early and go fishing every morning.
That makes more sense. Now, you can see that the tense of the "fishing" clause doesn't change, just because the context in which the fishing is done does. This sentence is also correct:
He said that after he had been out of school for five years he used to [not "had used to"] get up early and go fishing every morning.
However:
He said that after he had been out of school for five years he had gotten up early and gone fishing every morning.
Exactly why this is so, perhaps someone else can explain.
According to this document, https://www.englishclub.com/grammar/sentence/reported-speech-backshift.htm, "We NEVER use backshift when the original words are: past perfect, could, might, would, should"
– JavaLatte – 2016-08-01T17:35:49.657