In reality, in a conflict between the executive and legislative branches, the Courts are supposed to decide. If the alleged urgency to protect the 2020 election had any merit, the Courts would provide an expedited ruling, and the House could have had their witnesses if their demand had legal merit. The truth is this is all a cynical campaign ad for the November election, and the democrats are hoping their impeachment charade will do more to motivate their base than it does to alienate swing voters. Considering that they were going to lose in November, they might as well try this Hail Mary.
If the Chief Justice is currently presiding, my guess is he could, and would, expedite any appeal to the full Supreme Court.
I think there is a huge danger in letting Federal courts get involved in a dispute that relates to a process like impeachment that is explicitly laid out in the Constitution in terms of executive and legislative branch functions.