Judge Sullivan's reply is required by June 1, regardless of who writes it. Since there is no provision for a hearing or oral argument the decision will likely be made shortly after the first of the month.
You missed a key part of the discussion led by Sydney Powell herself.
The Appellate order was not to respond to the Writ motion. It was to explain the legal reasoning Sullivan is using in not dismissing the case.
In effect, the order requires Sullivan to lay his cards on the table.
In effect, the appellate court is giving Sullivan a chance to bow out by dismissing the case and thereby render the Writ matter moot.
Someone has decided the Judge will not bow out.
By responding June 1, the appellate court will take time to review what is surely expected to be a long response. Then they must ask Sullivan to respond to the motion for a Writ of Mandamus which triggers court rules of notice, calendar, reply etc.
By getting into the weeds of procedural delaytactics, the Deep State buys time for countermoves.