That sounds like a defendent of the record companies, not the executive branch trying to ensure law is faithfully executed. It would be one thing if the supreme court were ignoring laws, but arguing for the denial to review a case is hardly taking care that law is faithfully executed. Our executive branch is suggesting the subversion of due process in this case.
In other words, ignore the man behind the curtain.
Edit: to clarify- this is the final statement in the reasoning why due process should not be considered. Read the argument.
In the context of Constitutional challenges to Congressional laws, the DOJ has the same job. They will first seek to have review denied because that's the quickest way for Congress to win. If review is granted, the Solicitor General, whose office is part of the DOJ, will have to drag his ass to the Supreme Court and argue strenuously in favor of upholding the Constitutionality of Congress's law.
That's just the way the adversarial legal system works.
Also, your Due Process point is way out of bounds. Due Process does not entail the right to Supreme Court review. The Supreme Court could deny the petition for no other reason than "it sounded boring" and there would be no Due Process violation.