It's youtsubato, right? :)
そうにない seems treated as a single word/phrase in a dictionary, negative of そうだ. This site states そうに as 連用形 of そうだ. Seraching そうにない/そうではない (on this site or Google) suggests it is treated as a word in foreigner's grammar of Japanese.
連れていきそう = 連れていく + そうだ = likely to take to. So you are right on this.
Throughout the sentence, the object of 連れていく is omitted, and it is obviously Yotsuba the little girl. The subject of the second 連れていく is Koharuko the speaker.
- お兄ちゃんが連れていきそうにないところ : places where you (Tochan the father) are unlikely to take Yotsuba (to)
- ...に連れて行くのが私の役割かなと : I thought it would be my role to take Yotsuba to ....
So the whole sentence translates roughly as: "I thought it might be my duty to take Yotsuba somewhere you wouldn't take her to".
(The variation in vocab indicates just my unsureness.)