Can the Crown of Meereen chose attachments you already control to move them around?

Yep. It's best to take the text as literally as possible.

Response: After you win a challenge in which attached character participated, choose an attachment. Take control of that attachment, and attach it to an eligible card of your choice.

Says nothing about who controls the attachment when you choose it, so you can choose ANY attachment.

And since the word "then" doesn't appear either, the control change doesn't need to be successful for the moving part to resolve.