I don’t think that would work. What standing does the DOJ have to file such an unusual appeal?
Their standing is that they have the power to drop a case whenever there is entrapment, lack of evidence, coercion, prosecution misconduct, withholding of exculpatory evidence, etc., or any situation where the charges should of never been leveled.
In the case of Sullivan, DOJ can overstep Sullivan and go to the DC appeals court and ask for an order to stop the Amicus briefs fiasco.
Either way, if Sullivan goes rogue, he sets the case up for an immediate appeal/dismissal at the next upper level of court.
As far as I know, the DOJ has never come out and publicly stated in its court filings that there was any wrongdoing. Maybe that will change in the near future.