I think he doesn't have a choice
Functions of the Office of the Solicitor General
The major function of the Solicitor General's Office is to supervise and conduct government litigation in the United States Supreme Court. Virtually all such litigation is channeled through the Office of the Solicitor General and is actively conducted by the Office. The United States is involved in about two-thirds of all the cases the U.S. Supreme Court decides on the merits each year.
The Solicitor General determines the cases in which Supreme Court review will be sought by the government and the positions the government will take before the Court. The Office's staff attorneys participate in preparing the petitions, briefs, and other papers filed by the government in its Supreme Court litigation. The Solicitor General personally assigns the oral argument of government cases in the Supreme Court. Those cases not argued by the Solicitor General personally are assigned either to an attorney in the Office or to another government attorney. The vast majority of government cases are argued by the Solicitor General or by one of the Office's other attorneys.
Another function of the Office is to review all cases decided adversely to the government in the lower courts to determine whether they should be appealed and, if so, what position should be taken. The Solicitor General also determines whether the government will participate as an amicus curiae, or intervene, in cases in any appellate court.
Of course Ashcroft has a choice. He could resign.