- how can I know that 兄 in 【娘が「これ、借りて良い」と、兄に許可を求めている。】refers to 娘's elder brother rather than 筆者's elder brother?
In the first sentence the only subject is 娘 -- marked by が. Ergo, the one who receives permission to borrow from "brother" must be the daughter. And with that referent being set, then without a lot of other context, it's hard to make the brother anyone other than her own brother.
- if 兄 must refer to 娘's elder brother, why did 筆者 not write【娘が「これ、借りて良い」と、娘のお兄さんに許可を求めている。】? お兄さん is used here because 娘 becomes a third person from 筆者's point of view.
My sense as a non-native speaker is that on the simplest level without context, お兄さん and 兄 could both be used (with some other modifications for the sentence). But here 兄 should be strongly preferred.
First, one way of making it お兄さん would be to write from the daughter's perspective assuming she would refer to her brother as お兄さん rather than writing about the situation from the author's own perspective. The rest of the paragraph is not written that way but rather from the author's perspective about his children.
would be find here because that would make sense for her to say.
Second, the writing style here is appropriate for a Japanese essay or academic format. In such a context, there's no reason to use polite forms like お兄さん. Instead, the relations are stated simply. To give an example, 神様 is just written 神 in most academic writing.
Third, assuming the author is writing from his own experience, there's also a third reason. If this is the author's own family, then they are closer to the author than the audience. And you don't use honorifics about people who are closer to you than your audience.