I do not know whether DOJ has "Trump fatigue." But I would believe that it's prioritizing its investigations & prosecutions of Trump by their degree of overt threat to the Republic. We might not know whether it's already investigating Trump and Giuliani for selling pardons until one or more indictments is unsealed.

The Office of the Pardon Attorney, by the way, is in DOJ, not in the White House. So going directly to the White House to seek a pardon would keep the matter out of the ken of career DOJ lawyers who might well have objected.