An amendment to the Constitution to provide for an additional  constitutional office of:  “Vice President for the Minority”

This officer shall be determined from the second highest electoral votes for the office of President. 

The Vice President for the Minority shall control a “Minority Department of Justice” appointing a minority “Attorney General” and related. The budget of the "minority Department of Justice" shall be some modest percentage (example: 10%) of the main Department of Justice.  

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The Vice President for the Minority and his Minority Department of Justice shall have the authority and standing to bring cases against the US Government for the entire country where they believe the US Government has violated the constitution or rights of the people by the US Government's actions.

The Vice President for the Minority and the Minority Department of Justice shall have the authority and standing to defend the United States against lawsuits brought by the executive branch or others against the United States of America.

The Vice President for the Minority and the Minority Department of Justice shall have the authority and standing to bring suit against the President when there is evidence that the President, Vice President or other officer has accepted a prohibited emolument.

Cases brought by the Vice President for the Minority and the Minority Department of Justice shall be eligible for nationwide injunctive relief from the trial court level and through the all appellate levels.

The Vice President for the Minority shall have veto power on any firings by the President from Congressionally protected offices intended to be immune from political influence (such as, but not limited to, the Federal Reserve, Department of Justice US Attorneys). This Congressional intent shall be, at minimum, signified by the phrase "for cause" in the Congressional Act defining the office or agency.

The President shall have the power to veto similar firings by the Vice President for the Minority of appointments that the Vice President for the Minority has made.

The appointment vetoes above may be overridden by a 3/5 vote of the Senate.

The appointments of the Vice President for the Minority shall be subject to the same Congressional advise and consent rules as the appointments of the President. Author's self-brainstorming: Spirit of this good. But...! Congress already often slow-walks regular Presidential appointments. Need advise and consent rule for VPM, but, perhaps advise and consent of Senate minority, or, number of Senators equal to minority's representation only required for approval.

The Vice President for the Minority may appoint his own deputy. That deputy shall be his successor in the event of any disability, removal, etc.

The Vice President for the Minority and his deputy are not in the line of Presidential or Vice Presidential succession.

....all....of the "powers" here are "draft" in nature.  True constitutional scholars and lawyers would be needed to develop a final draft for constitutional amendment(s) and any related Congressional legislation.