Annabeth, if she were with Percy, would be more interested in the architecture and strategic planning of Troy. She could offer her knowledge to the Trojans or the Greeks, but they might not trust her either. And Percy would have to protect her from those who might see her as a threat or a sorceress.
The gods in Percy's world and the gods in the Trojan context would also interact in an interesting way. Zeus in both scenarios is a powerful figure. But the way they interact with mortals is different. In the fanfiction, there could be a showdown between the two versions of Zeus over who has more control over the events of the Trojan War, and Percy and his friends would be caught in the middle.