I don't think the dragons are choosing anyone. If they're both Targs, they'll be cool with both of them, just like they already seem to be.
This mission is idiotic though. Given what we just saw Dany/Drogon do to the Lannister forces, why not just have Dany/Drogon join the expedition right now. If it's all a bunch of fairy tales, she'll fly back home in five minutes. And if it's true, she can scorch the shit out of all the dead and get back to business in the fight against Cercei. Or does dragon fire not destroy the zombies, only dragonglass?
There's a lot of articles out today about how they're ****ing up the show with this breakneck pace. The big, meaningful reunions are just fast forwarded. The Jaime/Tyrion convo was garbage - barely anything was even said, and we didn't even see how the discussion ended. Just skip to Jaime going off to Cercei and Tyrion heading for the boat. Arya and Sansa have had two conversations, with the second one being contentious after years of being apart and having their family murdered, and just a few words to each other. No one even seems to be very impressed with Brans new abilities. Not worth talking to and probing the mind of the guy who knows everything that's ever happened to anyone? Dany and Jorah with a one minute surprise meet up, only for Jorah to go off on his little trip immediately after arriving, and looking perfectly good after being basically dead last week. Just not good.