You might already understand this, but let me check the basics again. First, the case particle that marks the subject and the object for 嫌い or 嫌になる is, either way, が. i.e. Aが Bが 嫌い (though this topicless sentence can't be a valid sentence without some conditions). When the subject is topicalized, it becomes Aは Bが 嫌い. When it comes to the object, Bは Aが 嫌い. When it happens to both, Aは Bは 嫌い.
Now, the reason the example uses は is, for one, because the sentences around the example one are already speaking of the woman(?) おさん and we want to indicate it as the topic, though how it uses その時 is a factor that inclines to use of が, on the other hand.
For another reason, I feel how the predicate stands for a negative feeling is perhaps a factor. This is something difficult to explain but sense of separating it with topicalization perhaps has something common with so-called contrastive は or how English speakers prefer "that" to "this" in a certain situation.
Edit: Reasons for topicalization are the same, even if multiple words are topicalized.
Indicates shared information (often called topical or thematic は): ◯◯さんは こういうのは どうですか？ → そうですね…私は こういうのは 嫌いです (In this reply, these repeated words tend to be omitted in practice.)*
Contrastive は: （△△さんは好きなようだが）私は…、（そういうのうはともかく）こういうのは…嫌い
Agrees to the negative predicate (a kind of the contrastive usage): This can be important when you highlight the word to negate among multiple words. e.g. 私は 昨日 そこには 行っていない vs 昨日は そこに… However, it's often used just because the predicate is negative even if there's only one word before it.
These factors are independent from (thus don't necessarily contradict with) each other. In the example case, if the questioner has asked someone else before hand and s/he has answered the opposite, or は is pronounced with stress, it's more likely to be interpreted as a contrastive usage. However, it still indicates the shared information. When it's neither really topical or contrastive, it tends to be explained with the remaining factor, the third one.
* I often hear a theory that claims there should be only one thematic は, but I wonder which should be the theme in this case.