Disclaimer: I have not finished watching Case Closed and I don't know much about the manga, so my information will be coming from solely his profile.
Can you provide further context to Conan's aim dodging feats? preferably a video/scan. I agree that she won't have the ability to instantly shoot Conan on sight, because he does have a few hiding spots, but assuming Shion is armed with an
AK-47, what hiding spots are there for Conan which could protect him from the weapon?
I disagree that Shion would underestimate Conan.
They both have knowledge of each other. And while Conan is smarter than Shion of course, she isn't a bumbling idiot, but a careful, methodical killer, who has the clear advantage in both physical stats and martial arts, on top of her enhanced senses which would be used to find a young boy in an otherwise empty, quiet school.
Conan does have his tranquiliser darts, and I think his fight plan is to hide and snipe her with it, using his glasses to make sure he can see Shion and not opting to go for his running shoes as getting up close to Shion would be suicide. You don't do that to someone who not only has an Avtomat Kalashnikova but also enhanced senses to see you coming yesterday.
An AK's range effective firing range is 350m, compared to several metres with Conan's darts.
For these reasons, I believe Shion Sonozaki takes this.