Interesting rule that I forgot about. A position player assigned to the minors must stay there for 10 days but can be brought back for a double header. So Seigler could be assigned to the minors on Thursday (replaced by Anthony), but brought back for the Yankees double header. I am guessing this is what they do. And then bring up Story on Sept 1 to start his "tryout". IKF stays down unless someone gets hurt. Probably add a pitcher as the 28th player.
Then you really don't have an issue until Mead is ready to come back.