Mistress Cogita says to visit Moonview Arena to find a pokémon who may have a Plate for you.
Go to the Coronet Highlands
After you have talked to Mistress Cogita about the crescent moon, during The Researcher of Myths, go to the Summit Camp at the Coronet Highlands. Approach Calaba and Melli.
Catch Cresselia
Cresselia is level 70. It creates a disorienting energy field in the area that reverses the movement controls, so press the stick in the opposite direction from the way that you want to go. When you get close enough, throw something at Cresselia from a distance so you won't get hit when it creates an energy field near itself, then throw a partner pokémon at it to start the battle. Reduce its HP and inflict status ailments to improve your chances of capturing it, then throw Ultra Balls (or whatever types of Poké Balls you have available) until you catch it.
When you have caught it, you get the Dread Plate.