If I remember correctly the ATE with Amarant and Freya appears once you completed the first round of the tournament.

However, I would suggest going back to your previous save if you didn't go to Dali before registering for the tournament. There you can enter the windmill and get a Cachusha and an Elixir. The Mayor's house contains the Mayor's Key, which will open the locked door in the windmill. Through there you'll find 30,000 gil and the Burman Coffee.