Strengthening Oversight of National Security Act of 2017 This bill amends the National Security Act of 1947 to add the Director of National Intelligence and the Chairman of the Joint Chiefs of Staff as statutorily required members of the National Security Council (NSC). (The current NSC members required by statute are the President, the Vice President, the Secretary of State, the Secretary of Defense, and the Secretary of Energy.) The bill prohibits the President from designating as additional members or attendees of the NSC any individuals in positions that are not subject to Senate confirmation (excluding the National Security Advisor and deputies, the Homeland Security Advisor and deputies, the Assistant to the President and Chief of Staff, the Counsel to the President, the Assistant to the President for Economic Policy, and the Assistant to the Vice President for National Security) unless Congress approves of the designation through enactment of a joint resolution or a bill. The cabinet-level interagency forum for policy consideration and decision-making that supports the NSC must reflect the membership of the NSC and any Senate-confirmed member of the executive branch designated by the President. Exceptions to this forum composition are permitted for a one-time decision-making action, but the President must notify Congress within 24 hours after execution of such an exception.